GIFT  OF 
Dr.   Horace  Ivle 


/     J 


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KOBOSON'S    MATHEMATICAL    SERIES. 


ELEMENTS 


PLANE  AND  SPHERICAL 


TRiaONOMETEY, 


WITH 


NUMEROUS  PRACTICAL  PROBLEMS. 


BY   ;     ; 

HORATIO    N.    ROBlfedN,    thiD 

AXJTHOB  OF  A  FULL  COtn^'OP  ilLflf^^llAlCil.' ,    .       , 


IVISON,  BLAKEMAN,  TAYLOR  &  CO., 

PUBLISHERS, 
NEW   YORK   AND   CHICAGO. 

1880. 


i( 


R-OEiisrsoisr's 


SERIES  OF  MATHEMATICS. 

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Mohinson's  Progressive  Table  Book, 

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Itobinson's  Progressive  Intellectual  JLritlitnetic,       -        -        .        . 

Jtobinson's  Jiudinients  of  Written  Arithmetic, 

Itobinson's  Progressive  Practical  Arithmetic, 

Itobinson's  Keg  to  Practical  Aritlttnetic,     -.---- 

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Itobinson's  Neic  Geometry  and  Trigonometry,  -        -        -        -       - 

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Jb' 


NOTICE. 


Upon  the  suggestion  of  many-  Teachers,  the  Publishers  have  thought 
best  to  bind  in  a  separate  volume  this  Treatise  on  Plane  and  Spherical 
Trigonometry ;  continuing,  however,  as  heretofore,  to  bind  it  up  with 
Robinson's  New  Geometry,  in  one  volume. 

In  this  form  it  will  bo  more  convenient,  and  less  expensive  for  those 
Teachers  and  Students  who  do  not  wish  to  take  up  the  Geometry  in  con- 
nection with  it,  or  who  desire  to  use  this  Treatise  on  Trigonometry  and 
some  other  author  on  Geometry. 


,a-    -! 


924227 


TKIGONOM-ETEY 


PAKT    I. 

PLANE    TRIGONOMETRY. 


SECTION  I. 

ELEMENTARY    PRINCIPLES. 

Trigonometry,  in  its  literal  and  restricted  sense,  has 
for  its  object  the  measurement  of  triangles.  When  it 
treats  of  plane  triangles  it  is  called  Plane  Trigonometry . 
In  a  more  enlarged  sense,  trigonometry  is  the  science 
which  investigates  the  relations  of  all  possible  arcs  of  the 
circumference  of  a  circle  to  certain  straight  lines,  termed 
trigonometrical  lines  or  circular  functions^  connected  with 
and  dependent  on  such  arcs,  and  the  relations  of  these 
trigonometrical  lines  to  each  other. 

The  measure  of  an  angle  is  the  arc  of  a  circle  inter- 
cepted between  the  two  lines  which  form  the  angle — the 
center  of  the  arc  always  being  at  the  point  where  the 
'.   two  lines  meet. 

:•  -..The  arc  is  measured  by  degrees^  minutes,  and  seconds; 
••  lli^re  being  360  degrees  to  the  whole  circle,  60  minutes 
\\\  fal'/one  degree,  and  60  seconds  in  one  minute.  Degrees, 
/..'  nlinutes,  and  seconds,  are  designated  by  °,  ',  ";  thus, 
%%\  27°  14'  21",  is  read  27  degrees  14  minutes  21  seconds. 
iV;.  .'The  circumferences  of  all  circles  contain  the  same 
•'•*••  number  of  degrees,  but  the  greater  the  radius  the  greater 
\-"l        *  (244) 


SECTION   I.  -  245 

is  tlie  absolute  length  of  a  degree.  The  circumference  of 
a  carriage  wheel,  the  circumference  of  the  earth,  or  the 
still  greater  and  indefinite  circumference  of  the  heavens, 
has  the  same  number  of  degrees ;  yet  the  same  number 
of  degrees  in  each  and  every  circumference  is  the  meas- 
ure of  precisely  the  same  angle. 

DEFINITIONS. 

1.  The  Complement  of  an  arc  is  90°  minus  the  arc. 

2.  The  Supplement  of  an  arc  is  180°  minus  the  arc. 

3.  The  Sine  of  an  angle,  or  of  an  arc,  is  a  line  drawn 
from  one  end  of  an  arc,  perpendicular  to  a  diameter 
drawn  through  the  other  end.  Thus,  BF  is  the  sine  of 
the  arc  AB^  and  also  of  the  arc  BDE,  BK  is  the  sine 
of  the  arc  BB. 

H 

4.  The  Cosine  of  an  arc  is  the  per-  j) ^^ 

pendicular  distance  from  the  center  of 
the  circle  to  the  sine  of  the  arc ;  or,  it  is 
the  same  in  magnitude  as  the  sine  of 
the  complement  of  the  arc.  Thus,  QF 
is  the  cosine  of  the  arc  AB\  but  0[F= 
KB^  which  is  the  sine  of  BB,  ^ 

5.  The  Tangent  of  an  arc  is  a  line  touching  the  circle 
in  one  extremity  of  the  arc,  and  continued  from  thence,  to 
meet  a  line  drawn  through  the  center  and  the  other  ex- 
tremity. Thus,  AH  is  the  tangent  to  the  arc  AB^  and 
BL  is  the  tangent  of  the  arc  BB, 

6.  The  Cotangent  of  an  arc  is  the  tangent  of  the  com- 
plement of  the  arc.  Thus,  DX,  which  is  the  tangent  of 
the  arc  BB^  is  the  cotangent  of  the  arc  AB. 

Kemark. — The  co  is  but  a  contraction  of  the  word  complement. 

7.  The  Secant  of  an  arc  is  a  line  drawn  from  the  center 
of  the  circle  to  the  extremity  of  the  tangent.  Thus,  CH 
is  the  secant  of  the  arc  AB^  or  of  its  supplement  BBE, 

8.  The  Cosecant  of  an  arc  is  the  secant  of  the  comple- 
ment.    Thus,  CZ,  the  secant  of  J?i>,  is  the  cosecant  oiAB. 

21* 


J 


246  PLANE    TRIGONOMETRY. 

9.  The  Versed  Sine  of  an  arc  is  the  distance  from  tho 
extremity  of  the  arc  to  the  foot  of  the  sine.  Thus,  A^ 
s  the  versed  sine  of  the  arc  AB^  and  DK  is  the  versed 
line  of  the  arc  BB, 

For  the  sake  of  brevity,  these  technical  terms  are  con- 
tracted thus :  for  sine  AB^  we  write  nn,  AB ;  for  cosine 
4  B,  T7e  write  coz,  AB;  for  tangent  AB,  we  write  ta7u 
AB,  etc. 

From  the  preceding  definitions  we  deduce  the  follow 
mg  obvious  consequences : 

1st.  That  when  the  arc  AB  becomes  insensibly  small, 
or  zero,  its  sine,  tangent,  and  versed  sine  are  also 
nothing,  and  its  secant  and  cosine  are  each  equal  to 
radius. 

2d.  The  sine  and  versed  sine  of  a  quadrant  are  each 
equal  to  the  radius ;  its  cosine  is  zero,  and  its  secant  and 
tangent  are  infinite. 

3d.  The  chord  of  an  arc  is  twice  the  sine  of  one  half 
the  arc.     Thus,  the  chord,  BG,  is  double  the  sine,  BF. 

4th.  The  versed  sine  is  equal  to  the  difi*erence  between 
the  radius  and  the  cosine. 

5th.  The  sine  and  cosine  of  any  arc  form  the  two  sides 
of  a  right-angled  triangle,  which  has  a  radius  for  its 
hypotenuse.  Thus,  OF  and  FB  are  the  two  sides  of  the 
right-angled  triangle,  CFB. 

Also,  the  radius  and  tangent  always  form  the  two 
sides  of  a  right-angled  triangle,  which  has  the  secant  of 
the  arc  for  its  hypotenuse.  This  we  observe  from  the 
right-angled  triangle,  OAIT, 

To  express  these  relations  analytically,  we  write 

sin.'  -f-  COS.'  =  i?  (1) 

R^     -I-  tan."  =  sec.'  (2) 

From  the  two  equiangular  triangles  CFB,  QAH^  we 

Jiave 

CF  ',  FB  ^  CA  I  AH. 


SECTION   I.  247 

That  id, 

COS.  :  sin.  =  B  :  tan. ;  whence,  tan.  «=  —  _  I       (3) 

cos 

Also,  OF  ,  CB  =  OA  :  CM. 

That  is, 

COS.  :  i2  =  jR  :  sec. ;  whence,  cos.  sec.  =  B^,   (4) 
The  tivo  equiangular  triangles,  QAH  and  ODL,  give 

CA  I  AH  =^  DL  :  DC. 

That  is, 

R  :  tan.  =  cot.  :  B\  whence,  tan.  cot.  =■  iT.     (5) 
Also,  CF  :  FB  =^  BL  :  BQ. 

That  is, 
COS.  :  sin.  =  cot.  :  i2;  whence,  cos.  B  =  sin.  cot.   (6) 
From  equations  (4)  and  (5),  we  have 

COS.  sec.  =  tan.  cot.  (7) 

Or,  COS.  :  tan.  =  cot.  :  sec. 

We  also  have     ver.  sin.  =  R —  cos.  (8) 

The  ratios  between  the  various  trigonometrical  lines 
are  always  the  same  for  arcs  of  the  same  number  of 
degrees,  whatever  be  the  length  of  the  radius ;  and  we 
may,  therefore,  assume  radius  of  any  length  to  suit  our 
convenience.  The  preceding  equations  will  be  more  con- 
cise, and  more  readily  applied,  by  making  the  radius 
equal  unity.  This  supposition  being  made,  we  have,  for 
equations  1  to  6,  inclusive, 

sin.'  +  cos.'  =1.  ( I ) 

1  +  tan.'  =  sec.»  (2) 

sin.      ,„,  1  ... 

tan.  = (3)  COS.  =   —  (4) 

COS.  sec. 

tan.  =  (5)  COS.  =  sin.  cot.     (6) 

cot. 

Let  the  circumference,  AEBH^  be  divided  into  fouj 
equal  parts  by  the  diameters,  AB  and  EH,  the  one  hori- 


248 


PLANE   TRIGONOMETRY. 


zontal  and  the  other  vert- 
ical. These  equal  parts 
are  called  quadrants,  and 
they  may  be  distinguished 
as  the  first,  second,  tldrd, 
and  fourth  quadrants. 

The  center  of  the  circle 
is  taken  as  the  origin  of 
distances,  or  the  zero  point, 
and  the  different  directions 
in  which  distances  are  esti- 
mated from  this  point  are  indicated  by  the  signs  +  and 
— .  1^  those  from  Q  to  the  right  be  marked  +,  those 
from  Q  to  the  left  must  be  marked  — ;  and  if  distances 
from  Q  upwards  be  considered  plus,  those  from  C  down- 
wards must  be  considered  minus. 

If  one  extremity  of  a  varying  arc  be  constantly  at  A, 
and  the  other  extremity  fall  successively  in  each  of  the 
several  quadrants,  we  may  readily  determine,  by  the 
above  rule,  the  algebraic  signs  of  the  sines  and  cosines 
of  all  arcs  from  0°  to  360°.  Now^,  since  all  other  trigo- 
nometrical lines  can  be-  expressed  in  terms  of  the  sine 
and  cosine,  it  follows  that  the  algebraic  signs  of  all  the 
circular  functions  result  from  those  of  the  sine  and 
cosine. 

We  shall  thus  find  for  arcs  terminating  in  the 

sin.  COS.  tan.  cot.  sec.  oosec        rera. 

1st  quadrant,  +         -|-         -f          -f-         +          -f  -f 

2d         "  +___        —        —         ^  4 

3d"  __4.4.__  + 

4th       "  —        +         —        —        +         —  -f. 


PROPOSITION   1. 

The  chord  of  60°  and  the  tangent  of  45°  are  each  equal  to 
radius  ;  the  sine  of  80°,  the  versed  sine  of  60°,  and  the  co* 
fiiie  of  60°  are  each  equal  to  one  half  the  radius* 


SECTION    1 


248 


With  (7  as  a  center,  and  CA  as  a 
ladius,  describe  the  arc  ABF,  and 
from  A  lay  off  the  arcs  AD  =  45°, 
AB  =  60°,  and  AE  =  90° ;  then 
is  EB  =  30°. 

1st.  The  side  of  a  regular  in- 
scribed hexagon  is  the  radius  of 
tLe  circle,  (Prob.  28,  B.  lY),  and  as  the  arc  subtended 
by  each  side  of  the  hexagon  contains  60°,  we  have  the 
chord  of  60°  equal  to  the  radius. 

2d.  The  triangle  CAR  is  right-angled  at  J.,  and  the 
angle  (7  is  equal  to  45°,  being  measured  by  the  arc  AD\ 
bence  the  angle  at  H  is  also  equal  to  45°,  and  the  trian- 
gle is  isosceles.  Therefore  AH  —  CA  —  radius  of  the 
circle. 

3d.  The  triangle  ABQ  is  isosceles,  and  Bn  is  a  per- 
pendicular from  the  vertex  upon  the  base ;  hence  An  = 
nC  =  Bm.  But  Bm  is  the  sine  of  the  arc  BE,  On  is  the 
cosine  of  the  arc  AB,  and  An  is  the  versed  sine  of  the 
same  arc,  and  each  is  equal  to  one  half  the  radius. 

Ilence  the  proposition  ;  the  chord  of  60°,  etc. 


PROPOSITION   II. 

Given,  the  sine  and  the  cosine  of  ttoo  arcs,  to  find  the  sine 
and  the  cosine  of  the  sum  and  of  the  difference  of  the  same 
arcs  expressed  by  the  sines  and  cosines  of  the  separate  arcs. 

Let  (r  be  the  center  of  the 
circle,  CD  the  greater  arc, 
and  DF  the  less,  and  denote 
these  arcs  by  a  and  h  re- 
spectively. 

Draw  the  radius  GD',  make 
the  arc  DE  equal  to  the  arc 
BF,  and  draw  the  chord  EF. 
From  i"  and  ^,  the  extremi- 
ties, and  J,  the  middle  point 


G   »r 


250  *     PLANE    TRIGONOMETRY. 

of  the  chord,  let  fall  the  perpendiculars  FM,  EP,  and 
IN,  ou  the  radius  GO.  Also  draw  DO,  the  sine  of  the 
arc  OB,  and  let  fall  the  perpendiculars  III  on  FM,  and 
^iTon  IK 

Now,  by  the  definition  of  sines  and  cosines,  BO  =» 
sin. a;  00  =  cos.a;  FI  =^  sin.6;  (7/=  cos.6.  We  are 
to  liud 

FM  =  sin.  (a  +  5) ;  GM  =  cos.  («  +  &);      - 
EP  =  sin.  {a  —  b)',  GP   =  cos.  {a  —  h). 

Because  IN  is  parallel  to  DO,  the  two  A*8,  GBO^ 
GIN,  are  equiangular  and  similar.  Also,  the  A  FHI  is 
similar  to  the  A  GIN-,  for  the  angles,  FIG  and  HIN, 
are  right  angles ;  from  these  two  equals,  taking  away  tKe 
common  angle  HIL,  we  have  the  angle  Fill—  the  angle 
GIN  The  angles  at  H  and  iVare  right  angles;  there- 
fore, the  A's  Fill,  GIN,  and  GBO,  are  equiangular 
and  similar;  and  the  side  HI  is  homologous  to  IN 
and  BO. 

Again,  as  FI  =  IE,  and  IK  is  parallel  to  FM, 

FH  =  IK,  and  HI  =  ^^. 
Bj  similar  triangles  we  have 

GB  :  BO  =   GI  :  IN 
That  is,  R  :  sin.a  =  co3.5  :  IN-,  or,  IN^  sin.ygsj^     ( 1 , 

Also,  GB  '.  GO  ^  FT  \  FH. 

That  is,  i2  :  cos.a  = «  ^m.h  :  HF;  or,  JPir=  cos^n^^     ^2) 

Also,  GB  :  GO  =   GI  :  GN 

That  is,  H :  cos.a  =  cos.  J :  GN;  or,  GN=^  ^y^-^^^^^     ( 3  j 

Also,  GB  :  DO  =  FI  :  IH. 

That  is,  i?  :  sin.ci  =  sin.6  :  IH;  or,  7ir=  ^l^hl^'l'j:!!.    (4) 

By  adding  the  first  and  second  of  these  equations,  we 
bave 

IN  -f  FH  =  F3I  -  sin.  (a  +  6). 


SECTION    I.  251 

a,,    ,  .           .     /     ,   7\       sin.a  C0S.6  +  cos.a  Bin.ft 
That  IS,       sin.  (a  -{-  b)  =^ ^ . 

By  subtracting  the  second  from  the  first,  since 

IN—  FJI=  ZZV—  iir=  UP,  we  have 

,         ,.        sin. a  C0S.5  —  cos.a  sin. J 
8in.(a  — J)= ^ 

J3y  subtracting  the  fourth  from  the  third,  we  have 

GN —  IH  =  GM  =  COS.  (a  4-  b)  for  the  first  member. 

TT                     /     .   IN       cos.a  C0S.6  —  sin. a  sin. 5      /.» 
Hence,     cos.  (a  +  6)  = .     (5) 

Mb 

13y  adding  the  third  and  fourth,  we  have 

GN-^-  in=  GN+NF=  (7P  =  co8.(a— 6). 

rr                 ,        TV      COS. «  COS. 5  4- sin. a  sin. i  ,ct\ 

Hence,  cos.(a  —  o)  — ~ .         ( o ) 

Collecting  these  four  expressions,  and  considering  the 
r»  lius  unity,  we  have  C'^^  ' 

sin. (a  +  J)  =  sin. a  cos.ft  +  cos.a  sin. J         ( 7 ) 

sin. (a — 6)=  sin. a  cos.6  —  cos.a  sin. 6         (8) 

cos.(a -\-h)  —  cos.a  cos.^  —  sin.a  sin.i         ( 9 ) 

.cos.(a  —  ^)  =  cos.a  cos.6  +  sin.a  sin.5       (10) 

Formuloe  {A)   accomplish  the  objects  of  the  proposi 
tion,  and  from  these  equations  many  useful  and  import 
ant  deductions  can  be  made.      The  following  are  the 
most  essential : 

By  a  Iding  ( "  )  to  ( 8 ),  we  have  ( n ) ;  subtracting  ( 8 ) 
fi\>m  ( 7  )  gives  ( 12 ).  Also,  ( 9 )  added  to  ( 10 )  gives  ( 13 ) , 
(  9  )  taken  from  ( 10 )  gives  ( 14 ). 

sin. (a -\-h)-\-  sin. (a — h)  =  2sin.a  cos.6  (  H  ) 

%\\\.(a-\-h)  —  sin. (a — 6)  =  2cos.a  sin. 6  (12) 

cos.(a  +  i)  +  cos.(a  —  h)  =  2cos.a  cos.6  ( 13 ) 

cos.(a  —  h) — cos.(a  +  b)  =  2sin.a  sin.6  (14) 

If  we  put  a-hb  =  A,  and  a  —  b  =  B,  then  (  H )  becomei. 
( 15 ),  ( 12 )  becomes  ( 16 ),  ( 13 )  becomes  ( 17  ),  and  ( 14 )  be 
comes  (18). 


w 


(B) 


252 


PLANF    TRIGONOMETRY. 


(0) 


rA-}'B\        /A— By 


Bm.A  +  sin.^  =  2sin.  (^-)  cos.  (^-^)      ( ^^1 
Bin.^-sm.5=2cos.(^±-?)sin.(^i:^)      (16) 

A  ^       A      ' 

'A^B\        /A  — By 


oos.A  +  COS.  J5  =  2cos.  (^  -)  cos.  (^-)     ( ^^ ) 
A-hB^ 


C08.J5  —  cos -4 


2 
^  — ^> 


^u^)""(^> ' 


18) 


sin. 


If  we  divide  ( 15 )  by  ( -6 ),  (observing  that  — '-  =  tan., 

COS. 

cos  1 

ttnd  -r-^  =  cot.  = as  we  learn  by  equations  ( 6 )  and 

sin.  tan. 


( 5 ),  we  shall  have 


Ai-B 


A—Bs 


A+By 


mn.A  +  sin 


.B  ''•'•(-^)  "'°^-(-2-)  ta°-(^-) 


sin.^  —  sin.i? 


Whence, 


A—B^ 


A—B^ 


(19) 


A-VBs 


A-By 


sin.JL+sin.^  :  sin.J.  — sin.5  =  tan.( \  :  tan.  ( ) 

V-     f  vV       ^  ^     2     ^ 

That  is :  The  sum  of  the  sines' (ff  cnty  two  arcs  is  to  the  dif- 
ference of  the  same  sines,  as  the  tangent  of  one  half  the  sum 
of  the  same  arcs  is  to  the  tangent  of  one  half  their  difference. 

By  operating  in  the  same  way  with  the  different  equa- 
tions in  formulae  ((7),  we  find, 

^sm.A  +  sin.B  /A-\-B\ 


1^) 


sin.^l  H-  sin.jB  /A  —  B\ 

cos.^  — cos^Z  ^  ^^**  V     2~~) 
sin.A — sin.^  ^A  —  B\ 

co8,A  Tco^  ==  *^^-  \     2~/ 
sin.J.— .sin..^  _  /^  +  A 

cos.^— cos.^  ""  ^^^-  \     2~/ 

cos.A  +  cos.^      ^^^'  \~^      / 


co&.B — COS.  J. 


tan. 


/A^B 


(20) 
(21) 
(22) 
(23) 

(24) 


SECTION    I.  253 

These  equations  are  all  true,  whatever  be  the  value 
of  the  arcs  designated  by  A  and  B ;  we  may,  therefore, 
assign  any  possible  value  to  either  of  them,  and  if  in 
equations  (20),  (21),  and  (24),  we  make  ^=  0,  we  shall 
have, 

Bm.A         J.      A          1  ,ncx 

=  tan.  —  = —I  ( 25 ) 


i  -f  cos.^  2      cot^A 

Bm,A              .A           1  f^^. 

=  cot.  -^  = — T  ( 26 ) 


(g  1-— cos.^  2      tan.J^ 

1 4-  cos.^      cot.^^  1 


(27) 


1— cos.Jl      tan.J^      tan'l^ 
J£  we  now  turn  back  to  formulae  (A),  and  divide  equa- 
tion (7)  by  (9),  and  (8)  by  (10),  observing  at  the  same 

time  that  — '-  =  tan.,  we  shall  have, 
cos.  ' 

+«    /     ,  i\      sin.a  cos.  J  4-  cos.a  sin. 6 
tan. (a  -f  6)  = 


tan. (a — 5)  = 


cos.a  C0S.6 — sin. a  sin.b 
sin. a  cos. 5  —  cos.a  sin. 5 


cos.a  C0S.6  +  sin. a  sin. 6 

By  dividing  the  numerators  and  denominators  of  the 

second  members  of  these  equations  by  (cos.a  cos.i),  we 

find, 

sin. a  C0S.5    cos.a  sin. 6 

,      ,.     cos.a  C0S.6    cos.a  C0S.6      tan.a-ftan.ft 

tan.(a4-5)= , -. -. — ==- — (28) 

cos.a  COS. 0    sm.a  sin.6     1 — tan. a  tan. o 

cos.a  COS. 6    cos.a  COS. 5 
sin. a  COS. 5    cos.a  sin.6 


,      ,,    cos.a  cos.6    cos.a  COS.6      tan. a — tan. 5 

tau.(a— 5)= -. — . .     ,==-— ^ -— -     (29) 

^        '    cos.a  C0S.6    sin. a  sin. D    l+tan.a  tan.6 

cos.a  COS. 6    cos.a  C0S.6 
If  in  equation  (H),  formulae  (B),  we  make  a  =  6,  we 
ihall  have, 

sin. 2a  =  2sin.a  cos.a         (30) 

Making  the  same  hypothesis  in  equation  (13),  gives, 
co8.2a  +  1=  2co8'.a         (31) 
22 


254  PLANE    TRIGONOMETRY. 

The  same  hypothesis  reduces  equation  (14)  to 

l_cos.2a  =  2siIl^a         (32) 

The  same  hypothesis  reduces  equation  ( 28 )  to 

,       o  2tan.a  ,oq\ 

tan.2a  =  - — —  -i-  (3^) 

1  —  tsijr.a 

If  we  substitute  a  for  2a  in  (31)  and  (32)^  we  shall  hav6 
1  +  cos.a  =  2cos.'ia.         (34) 
and  1 —  cos.a  ==  2sin.'Ja.         (35)   ^ 


PROPOSITION   III. 

In  any  right-angled  plane  triangle^  we  may  have  the  fol- 
lowing proportions : 

1st.  The  hypotenuse  is  to  either  side,  as  the  radius  is  to  the 
sine  of  the  angle  opposite  to  that  side,^ 

2d.  One  side  is  to  the  other  side,  as  the  radius  is  to  the  tan- 
gent of  the  angle  adjacent  to  the  first  side, 

3d.  One  side  is  to  the  hypotenuse,  as  the  radius  is  to  the 
eecant  of  the  angle  adjacent  to  that  side. 

Let  CAB  represent  any  right- 
angled  triangle,  right-angled  at 
A. 

(Here,  and  in  all  cases  hereafter,  we  shall  represent  the  angles  of  a 
triangle  by  the  large  letters  A,  B,  C,  and  the  sides  opposite  to  them, 
by  the  small  letters  a,  6,  c.) 

From  either  acute  angle,  as  C,  take  any  distance,  as 
CB,  greater  or  less  than  CB,  and  describe  the  arc  BE. 
This  arc  measures  the  angle  0.  From  B,  draw  BF  par- 
allel to  BA ;  and  from  JE,  draw  EGi,  also  parallel  to  BA 
or  BF, 

By  the  definitions  of  sines,  tangents,  secants,  etc,  BF 
is  the  sine  of  the  angle  0;  EGi  is  the  tangent,  CO  the 
secant,  and  CF  the  cosme. 


SECTION    I. 


255 


Now,  by  proportional  triangles  we  have, 

CB  :  BA=  CD:  DF     or,  a  :  c  =  R  :  sm.O 
CA:AB=CU:  EG    or,  b  :  c 
CAiCB  ^  QE :  CG 

Hence  the  proposition. 

ScnoLiuM. — If  the  hypotenuse  of  a  triangle  is  made  radius,  one  sidfl 
is  the  sine  of  the  angle  opposite  to  it,  and  the  other  side  is  the  cosine 
of  the  same  angle.     This  is  obvious  from  the  triangle  CDF» 


R  :tan.(7 
o\\  h  :  a  =  R  :  sec.  (7 


PROPOSITION    IV. 

In  any  triangle,  the  sines  of  the  angles  are  to  one  another 
08  the  sides  opposite  to  them. 

Let  ABC  be  any  tri- 
angle. From  the  points 
A  and  B,  as  centers, 
with  any  radius,  de- 
Hcribe  the  arcs  meas- 
uring these  angles,  and 
draw  pa,  CD,  and  mn, 
perpendicular  to  AB. 

Then,  pa  =  sin.^,  and  mn  =  sin.^. 

By  the  similar  A's,  Apa  and  ACD,  we  have, 
R  :  Bin. A  =  biCB;  or,  R{CI))  =  b  sin.A 

By  the  similar  a's,  Bmn  and  BCD,  we  have, 

R  :  sm.B  =  a:  CD;  or,  R(CD)  =  a  sm.B     (2) 

By  equating  the  second  members  of  equations  ( 1 ) 
and  (2) 

b  sin, A  =  a  sin. 5. 

Uence,  sin.^  :  sin.^  =  a  :  b 

Or,  a  :  b  =  sin.A  :  sin.^. 

ScnoLiUM  1. — When  either  angle  is  90°,  its  sine  is  radius. 

Scholium  2. — When  CB  is  less  than  AC,  and  the  angle  B,  acute, 
the  triangle  is  represented  by  ACB.  When  the  angle  B  becomes  B^, 
it  is  obtuse,  and  the  triangle  is  A  CB^ ;  but  the  proportion  is  equally 


(1) 


256  PLANE    TRIGONOMETRY. 

true  with  either  triangle ;  for  the  angle  CB^D  =  CBAy  and  the  8iu< 
of  CB^D  is  the  same  as  the  sine  of  AB^C.  In  practice  we  can  deter- 
mine which  of  these  triangles  is  proposed,  by  the  side  AB  being 
greater  or  less  than  AC',  or,  by  the  angle  at  the  vertex  C  being  large, 
as  A  CB,  or  small,  as  A  CB^. 

In  the  solitary  case  in  which  AC,  CB,  and  the  angle  A,  are  given, 
and  CB  less  than  AC,  we  can  determine  both  of  the  A's  A(^B  and 
A  CB'' ;  and  then  we  surely  have  the  right  one. 


PROPOSITION    V. 

If  from  any  angle  of  a  triangle,  a  perpendicular  he  let  fall 
on  the  opposite  side,  or  base,  the  tangents  of  the  segments  of 
the  angle  are  to  each  other  as  the  segments  of  the  base. 

Let  J.^ (7  be  the  triangle.  Let  fall 
the  perpendicular  CB,  on  the  side 
AB, 

Take  any  radius,  as  Cn,  and  de- 
scribe the  arc  which  measures   the      A.      QV^^^^^B 
angle  C,     From  n,  draw  qnp  parallel  to  AB.     Then  it  is 
obvious  that  np  is  the  tangent  of  the  angle  BOB,  and  nq 
is  the  tangent  of  the  angle  AQD. 

Now,  by  reason  of  the  parallels  AB  and  qp,  we  have, 
qn  :  np  =  AB  :  BB 

That  is,         tan.^OZ)  :  \s.xi,BQB  =  AB  :  BB. 


PROPOSITION   VI. 

If  a  perpendicular  be  let  fall  from  any  angle  of  a  triangh 
to  its  opposite  side  or  base,  this  base  is  to  the  sum  of  the  other 
two  sides,  as  the  difference  of  the  sides  is  to  the  difference  of 
the  segments  of  the  base, 

(See  figure  to  Proposition  5.) 

Let  AB  be  the  base,  and  from  0,  as  a  center,  with  the 
shorter  side  as  radius,  describe  the  circle,  cuttiiig  AB  in 
Q,  and  AOm  F-,  produce  AQ to  E. 


SECTION    I.  257 

It  is  obvious  that  AF  is  the  sum  of  the  sides  AO  and 
CB,  and  AF  is  their  difference. 

-Also,  AD  is  one  segment  of  the  base  made  by  the  per- 
pendicular, and  BD  =  BCr  is  the  other;  therefore,  the 
difference  of  the  segments  is  AG. 

Aa  A  is  a  point  without  a  circle,  by  Cor.  Th.  18,  B. 
in,  we  have 

AE  X  AF  =  AB  X  AG 

Hence,  AB   :   AF  =  AF  :   AG. 

PROPOSITION   VII. 

TJie  sum  of  any  two  sides  of  a  triangle  is  to  their  difference^ 
as  the  tangent  of  one  half  the  sum  of  the  angles  opposite  to 
these  sides,  is  to  the  tangent  of  one  half  their  difference. 

Let    ABQ   be    any   plane    triangle. 
Then,  by  Proposition  4,  we  have, 

BQ:  AQ^  sin.J.  :  Qin.B. 


Hence,  A  B 

«C  +  ^(7:^C— ^(7=sin.^-fsiii.^:sin.^— sin.^(Tli.9,B.II> 
But, 

tan.  ( — i —  j  :  tan.  ( — - —  \  =  sm.A  4-  sin.^  :  sin.-i 

—  sin.^,  (eq.  (19),  Trig.) 

Comparing  the  two  latter  proportions,  (Th.  &,  B.  U), 
we  have, 

^C'+ ^(7:5C— ^C=~tan.  (^4-^^  :tan.  (^1:=1^) 
Hence  the  proposition. 

PROPOSITION   VIII. 

Given,  the  three  sides  of  any  plane  triangle,  to  find  some 
relation  which  they  must  hear  to  the  sines  and  cosines  of  thi 
respective  angles. 

22*  R 


258 


PLANE    TRIGONOMETRY. 


Let  ABC  be 
the  triangle,  and 
let  the  perpen- 
dic  alar  fall  either 
upon,  or  without 

thebase,  as  shown    c  _  _ 

m    the     ngures. 
By  recurring  to  Th.  41,  B.  I,  we  shall  find 


c     p 


CD  = 


c* 


2a 


(1) 


Kow,  by  Proposition  3,  we  have 

B  :  COS.  0  =  b  :  CD. 


Therefore, 


nj)  _   ^  COS.  0 

R 


(2) 


Equating  these  two  values  of  CD,  and  reducing,  we 
Have 

COS.  (7  =  —  ^ 


2ah 


Arn) 


In  this  expression  we  observe,  that  the  part  <?,  whose 
square  is  found  in  the  numerator  with  the  minus  sign,  is 
Uie  side  opposite  to  the  angle ;  and  that  the  denominator 
is  twice  the  rectangle  of  the  sides  adjacent  to  the  angle. 
From  these  observations  we  at  once  draw  the  following 
expressions  for  the  cosine  A,  and  cosine  B : 


COS.  A  — 


R(h'  +  c'  —  a' 


cos 


2ba 


in) 


(i>) 


:ac 


As  these  expressions  are  not  convenient  for  logarith- 
mic computation,  we  modify  them  as  follows: 
If  we  put  2a  =  A,  in  equation  ( 31 ),  we  have 

COS.  A  +  1  =  2cos.'  J  J.. 

In  the  preceding  expression,  (w),  if  we  consider  radius 
ttxiity,  and  add  1  to  both  members,  we  shall  have 


COS.  J.  +  1  =  1  H- 
Therefore,      2cos.'J^  = 


SECTION  I.  259 

J'  -f  (;>  —  a« 


2hc 
26c  4- J' +  <?'  —  «• 


2bc 

^  (h  +  g)'  —  g' 
26(? 
CoDsidering  6  +  c  as  one  quantity,  and  observing  that 
(5  +  c)' — a'  is  the  difference  of  two  squares^  we  have 
(t-|.c)«— a»=(5+c-|-a)  {l-irc—a) ;  but  (?»+c— a)=Z>4-c+a— 2a. 
Hence,      2cos.'JA  =  (&  +  <^  +  a)  (^^+ <^  +  ^  -  2a). 

n  « ij         ^ 2 )  V 2—       "";. 

Or,     cos.'J^  =  J 

0(7 

By  putting =  «,  and  extracting  square  root, 

the  final  result  for  radius  unity  is 

COS.  U  =  sj'-^^. 

^  DC 

For  any  other  radius  we  must  write 

C08.W    =    J I^'' (>-"). 

^  DC 

By  inference,   cos.  IB  =  v/^'^  (^  ~  ^^. 

AlBO,  COS.  i(7  =  Ay^MfZI). 

In  every  triangle,  the  sura  of  the  three  angles  is  equal 
to  180° ;  and  if  one  of  the  angles  is  small,  the  other 
two  must  be  comparatively  large ;  if  two  of  them  are 
email,  the  third  one  must  be  large.  The  greater  angle 
is  always  opposite  the  greater  side;  hence,  by  merely 
inspecting  the  given  sides,  any  person  can  decide  at 
once  which  is  the  greater  angle;  and  of  the  three  pre- 


260  PLANE    TRIGONOMETRY. 

ceding  equations,  tliat  one  should  be  taken  which  apphca 
to  the  greater  angle,  whether  that  he  the  particular 
angle  required  or  not ;  because  the  equations  bring  out 
the  cosines  to  the  angles ;  and  the  cosines  to  very  small 
arcs  vary  so  slowly,  that  it  may  be  impossible  to  decide, 
with  sufficient  numerical  accuracy,  to  what  particular 
arc  the  cosine  belongs.  For  instance,  the  cosine  0.999999, 
carried  to  the  table,  applies  to  several  arcs;  and,  of 
course,  we  should  not  know  which  one  to  take ;  but  this 
difficulty  does  not  exist  when  the  angle  is  large ;  there- 
fore, compute  the  largest  angle  first,  and  then  compute 
the  other  angles  by  Proposition  4. 

But  we  can  deduce  an  expression  for  the  sine  of  any 
of  the  angles,  as  well  as  the  cosine.  It  is  done  as  fol- 
lows: 

EQUATIONS  FOR   THE   SINES  OF  THE   ANGLES. 

Besuming  equation  ( wi ),  and  considering  radius  unity, 
we  have 

a»  +  h'  —  (?» 

COS.  a  =  —^^3—. 

Subtracting  each  member  of  this  equation  from  unity, 
gives 

l_eos.a=l-(fl±|^).        (1) 

Make  2a  =  (7,  in  equation  (32)  j  then  a  =  ^0, 
and    1  —  COS.  (7  =  2sin.4(7.  (2) 

Kv^uating  the  second  members  of  (1)  and  (2), 
2ab  —  a'  —  6»  -f  c» 


2sin.»J(7  = 


2ab 


_  e'-{a-  bf 
2ab 

(c  -{-  b  —  a){c  -\-  a  —  b) 
*  2^ 


SECTION   I.  261 

/c  +  b  —  a\   /c  +  a  —  h\ 

Or,  Bin.4(7  =  iZIUlL^ZAZZ. 

ab 

BuL—^~=  — ^ a,  and---  =  — ^ 1 

Put       ^ =  «,  as  before;  then, 

^  ab 

By  taking  equation  {p ),  and  proceeding  in  the  same 
manner,  we  have 

ac 

From  (n),  sin.iA  =   \/EEK5jE^. 
^  cb 

The  preceding  results  are  for  radius  unity ;  for  any 
other  radius,  we  must  multiply  by  the  number  of  unita 
in  such  radius.  For  the  radius  of  the  tables  we  write 
R;  and  if  we  put  it  under  the  radical  sign,  we  must 
write  R^;  hence,  for  the  sines  corresponding  with  our 
loejarithmic  table,  we  must  write  the  equations  thus, 

8in.jA=  ^MESEEZ). 

^  be 

^  ac 

^  ab 

A  large  angle  should  not  be  determined  by  these 
equations,  for  the  same  reason  that  a  small  angle  should 
not  be  determined  from  an  equation  expressing  the 
cosine.  • 

In  practice,  the  equations  for  cosine  are  more  gener- 
ally used,  because  more  easily  applied. 


262  PLANE   TRIGONOMETRY. 

The  formulae  which  we  have  thus  analytically  devel- 
oped,  express  nearly  all  the  important  relations  between 
the  sines,  cosines,  and  tangents  of  arcs  or  angles;  and 
we  have  also  demonstrated  all  the  theorems  required  for 
the  determination  of  the  unknown  parts  of  any  plane 
triangle,  three  of  the  parts  of  which  are  given,  one  at 
least  being  a  side. 

Such  relations  might  be  indefinitely  multiplied,  but 
those  already  established  are  »^ufficient  for  most  practical 
purposes,  and  when  others  are  required,  no  difficulty 
will  be  found  in  deducing  them  from  these. 

The  following  geometrical  demonstrations  of  many  of 
the  preceding  relations,  are  oflered,  in  the  belief  that 
they  will  prove  useful  disciplinary  exercises  to  the  stu- 
dent. > 
1st.  Let  the  arc  AD  =A ;  then  I)G-= sin.  A ;  CGr=zcoa,A , 
i>J=sin.iJ.;  J.i)=2sin.JJL;  CI=co3.iA; 
CI=I)0;  and  i>^=2i>0=2cos.JX 
The  angle,  DBA,  is  measured  by 
one  half  the  arc  AD ;  that  is,  by  J  J.. 
Also,        ADa  =  DBA  =  iA, 
Now,  in  the  triangle,  BDCr,  we  have 

sm.DBa  :  i>(7=sin.90°  :  BD. 
That  is,  sin.J^ :  sin.J.=l :  2cos.JJ.. 
Or,  sin.J.=2sin.JJ.  cos.}^; 

which  corresponds  to  equation  ( 30 ). 

In  the  same  triangle, 
Bin.9d°  .  BD==am.BDa  :  BG;  and  sm.BDa^coa.DBG 
That  is,  1  :  2cos.JJ.=co8.JJ.  :  l+cos.J.. 
Or,  2co8.' J^=l-f-cos.J[,  same  as  equation  (34). 

In  the  triangle,  DGA,  we  have, 

Bin.90°  j  ^i>  =  8in.G^i)^  :  a  A, 
That  is,  1  :  2sin.  J^  =  sin.  J  J.  :  1  — cos.J.. 
Or,  2sin.''  ^A  =  1 —cos.  J.,  same  as  equation  ( 36  X 


SECTION    I, 


268 


By  similar  triangles,  we  have, 

BAiAD^ABiAa,- 

That  is,  2  :  2sin.J^  =  2sin.Jj4  :  versed  sin. J.. 

Or,  versed  sin. JL  =  2sin.'  ^A, 

2d.  From  C  as  the  center,  with  CA  as  the  radiu^ 
describe  a  circle.  Take  any  arc, 
ABj  and  call  it  A  ;  and  AD  a  less 
arc,  and  call  it  ^ ;  then  JBI)  is  the 
difference  of  the  two  arcs,  and  must 
be  designated  by  (A — B) ;  arc  AG 
K=  arc  AB ;  therefore, 

£iroI)a=-A  +  B;  Ua==sm.A; 

£n  =  8m.B;  6^n=:sin.J.  + sin.^; 
Bn  =  sin.^  —  sin.^. 
Fm  =  ml)  =  CH=  cos.B ;  mn  —  cos. J. ; 
therefore,   Fm  +  mn  =  cos.^  +  cos.^  =  Fn ; 
mD  —  mn  =  cos.^  —  cos.^  =  nD ; 

and  i)6^  =  2sin.(^-±^). 

Because,  NF-^AB)  AB-^ NF=-A-^ B\ 

therefore,  180°  —  (^  +  j5)  =  arc  FB^ 


or. 


90°-(^4^)=iarcZfi. 


But  the  chord,  FB,  is  twice  the  sine  of  J  arc  FB ; 


that 


A  +  B. 


is,      FB  =  2sin.  (90°  - 1     )  =  2cos.  {~^\ 
The  L_w(ri)  =  [_BFB,  because  both  are  measured 

by  one  half  of  the  arc  BB;  that  is,  by  ( — - — V  and  the 

two  triangles,  GnB  and  FnB,  are  similar. 
The  angle,  GFn,  is  measured  by  ( — - — \ 
In  the  triangle,  FBG,  Fn  is  drawn  from  an  angle  per 


264  PLANE    TRIGONOMETRY. 

pendicular  to  the  opposite  side ;  therefore,  by  Propositioi* 
5,  we  have, 

an  :  nB^tsin.aFn  :  tsin.BFn, 


A-{-B^ 


That  is,  8iu.J.+sin.^  :  sin.  J. — sin.J5=tan.( — — — ")  : 


tan. 


{^^-j^y     This  is  equation  ( 19 ).  ^J^' 

In  the  triangle,  GnD,  we  have, 

sin. 90°  :  I)Gr  =  &m.nI)G  :  Crn;  sm.nBG-^cos.nCrl), 

That  is,  1  :  2sin.  (f^)  =  cos.  (^^)  :  sin.  JL+sin.^. 

Or,  sin.^  +  sin. J5  =  2sin.  (^y^)  cos.  {^-^)> 

the  same  as  equation  (15). 
3d.  In  the  triangle,  FnB,  we  have, 

sin.90  :  FB  =  sin. BFn  :  Bn. 

That  is,  1  :  2cos.(^±^)  =  ^m.{^?-)  :  sin.^^sin.i^. 

Or,  sin.^-sin.^  =  2cos.  {^~)  sin.  (^~^), 

the  same  as  equation  (16). 
4th.  In  the  triangle,  FBn,  we  have, 

sin.90  :  FB  =  cos.BFn  :  Fn. 

That  is,  1  :  2cos.  {^~^)  =  cos.(^^i:p?)  :  cos.^+cos.J9. 

Or,  cos.^  +  cos.^  =  2cos.  (f^^)  cos.  (^  ~^),  the 
«ame  as  equation  ( 17 ). 

5th.  In  the  triangle,  GnB,  we  have, 

sin.90°  :  ai)  =  ain.nai)  :  nJ), 

That  is,  1  :  2sin.  (-^— )  =  sin.  (-^-  )  :  C08.5— cos.^ 
*.he  same  as  equation  ( 18 ). 

6th.  In  the  triangle,  FGrn,  we  have, 

sin.  GrFn  :  G-n  =  cos.  GFn  :  Fn. 


SECTION    I.  265 

That  is,  sin.     ^     :  8in.J.+sin.^  =  cos.  —^~—  :  cos.A-^ 
oos.B, 

Or,  (sin.^  +  sin.j^)  cos.  { — - — )  =  (co8.-4.  +  coe.J?)  sin. 
A-{B 


r-¥)- 


.     A  +  B 

sin. 


cos.^ — 2 — 

same  as  equation  (20). 

7tli.  In  the  triangle,  FnB,  we  have, 

Fn  :  nB  : :  1  :  tau.BFn, 

Thatis,  C08.J5  +  C0S.J.  :  sin.JL— sin.jB  ::  1  :  tan.J(A — B)» 

^  &m.A — sin.5      ,       /A — B\     ,, 

Or,  — i -  =  tan.  ( — - — ),   the    same 

'  cos.  J- -f  COS..B  \     2     r 

as  equation  (22). 

8th.  In  the  triangle,  GnB,  we  have, 

On  :  nD  ::  1  :  tan.wG^i). 
That  is, 

— o —  )» 

cos.  B  —  COS.  A        ,       /A  —  B\ 

or,  — i J—   -r.  =  tan.  ( — - —  ). 

'  sin.  A  +  sin.  B  \      2      / 


NATURAL    SINES,    COSINES,    ETC. 

When  the  radius  of  the  circle  is  taken  as  the  unit  of 
measure,  the  numerical  values  of  the  trigonometrical 
lines  belonging  to  the  different  arcs  of  the  quadrant,  bo- 
come  natural  sines,  cosines,  etc.  They  are  then,  in  fact, 
but  numbers  expressing  the  number  of  times  that  these 
line^  contain  the  radius  of  the  circle  in  which  they  are 
taken.  The  tables  usually  contain  only  the  sines  and 
cosines,  because  tliese  are  generally  sufficient  for  practi- 
§3 


266  PLANE    TRIGONOMEIR  Y. 

cal  pui-poses,  and  the  others,  when  required,  arc  rnailily 
expressed  in  terms  of  them. 

We  proceed  to  explain  a  method  for  computing  a  table 
of  natural  sines  and  cosines. 

It  was  shown,  in  Book  Y,  that  the  linear  value  of  thci 
arc  180°,  in  a  circle  whose  radius  is  unity,  is 

3.141592653. 

This  divided  by  180  x  60,  the  number  of  minutes  id 
180°,  will  give  the  length  of  one  minute  of  arc,  which  ia 

.00029088820867. 

But  there  can  be  no  sensible  difference  between  the 
length  of  the  arc  V  and  its  sine ;  and,  within  narrow 
limits,  that  sine  will  increase  directly  with  the  arc. 


Hence, 

sin. 

1'  = 

.0002908882. 

sin. 

2'  = 

.0005817764. 

sin. 

3'  = 

.0008726646. 

sin. 

4'  = 

.0011635528. 

sin. 

5'  = 

.0014544410. 

sin. 

6'  = 

.0017453292. 

sin. 

7'  = 

.0020362175. 

sin. 

8'  = 

.0023271057. 

sin. 

9'  = 

.0026179939. 

sin. 

10'  = 

.0029088821. 

Beyond  this,  the  error  which  would  arise  from  taking 
the  arc  for  its  sine,  upon  which  the  above  proceeds, 
would  affect  the  final  decimal  figures;  and  we  must, 
therefore,  continue  the  computation  of  the  series  by 
other  processes.  To  find  the  values  of  the  cosines  of 
arcs,  from  1'  to  10',  we  have 

cos.  =   ^1  —  sin.^  =  1  —  J  sin.*,  nearly. 

That  is,  when  the  sines  are  very  small  fractions,  an  ih 
the  case  for  all  arcs  below  10',  we  can  find  the  cosine  htf 
subtracting  one  half  of  the  square  of  the  sine  from  unity. 


SECTION    I.  267 

WlieDce,        COS.  1'  =  .9999999577. 

COS.  2'  =  .9999998308. 

COS.  3'  =  .9999993204. 

COS.  4'  =  .99999932304. 

COS.  5'  •=  .99999894290. 

COS.  6'  =  .99999847753. 

COS.  7'  =  .99999792735. 

COS.  8'  =  .9999973035. 

COS.  9'  =  .9999965730. 
COS.  10'  =  .9999957703. 

The  natural  sines  of  arcs,  differing  by  1',  from  10  up 
lo  1°,  raaj  be  computed  from  those  of  arcs  less  i  ad 
10',  oy  means  of  equation  ( H ),  group  B,  which  is 

sin.  (a  -{■  h)  =  23in.  a  cos.  b  —  sin.  {a  —  b); 

And  when  a  =  b,  this  equation  becomes 

sin.  2a  =  2sin.a  cos.  a.     Eq.  (30). 

To  find  the  sine  of  11',  we  make  a  =  6',  and  6  =   5'; 

then  sin.  11'  =  2sin.  6'  cos.  5'— sin.  1'==  .00319976^13 

a  =  6  =  6',       sin.  12'  =  2sin.  6'  cos.  6'. 
a  =  7',  i  =  6',  sin.  13'  =  2sin.  7'  cos.  6'  —  sin.  V. 
a=b=:7,       sin.  14'  =  2sin.  7'  cos.  7'. 
a  =  8,  6  =  7,    sin.  15'  =  2sin.  8'  cos.  7'  —  sin.  V 

And  so  on  to  the 

sin.  30'  =  2sin.  15'cos.l5'. 
sin.l°  =  sin.  60'  =  2sin.  30'cos.30'. 
sin.  2°   =  2sin.l°  cos.  1°. 
sin.  3°   =  2sin.  2°  cos.  1*"  —  sin.  1°,  etc.,  etc.,  etc. 

This  process  may  be  continued  until  we  have  found 
ll:c  sires  and  cosinea  of  all  arcs  differing  by  1',  from  0 
to  90°,  the  values  of  the  cosines  being  deduced  success- 
ively from  those  of  the  sines  by  means  of  the  formula. 


COS.  =   v/l  —  sin.'. 
In  this  calculation,  we  began  by  assuming  that,  for 
small  arcs,  Ihe  sines  and  the  arcs  were  sensibly  equal. 


268  PLANE    TRIGONOMETRY. 

[t  must  be  remembered  that  this  is  but  an  approxiina. 
tion ;  and  although  the  error  in  the  early  stages  of  the 
process  is  not  sufficient  to  affect  any  of  the  decimal  lig- 
ures  which  enter  the  tables,  it  will  finally  become  so, 
since  it  is  constantly  increased  in  the  operations  by 
which  the  sines  and  cosines  of  the  larger  arcs  are  de- 
duced from  those  of  the  smaller.  When  the  error  has 
been  thus  increased  until  it  reaches  the  order  of  the  last 
decimal  unit  of  the  table,  which  assigns  our  limit  of 
error,  we  must  have  the  means  of  detecting  and  correct 
ing  it. 

Thij^  consists  in  calculating  the  sines  and  cosines  of 
certain  arcs  by  independent  processes,  and  comparing 
them  vTith  those  found  by  the  above  method. 

We  have  seen,  for  example,  (Prop.  7,  B.  V),  that  the 
chord  of 

80^^    517638090 ;  whence,  sin.  15°  =.258819045. 

15"  :r=  .2610523842;       «       "       7°  30'       =.130526192. 
7^30'  ^  .1308062583;       "       "      3°  45'       =.0654031291. 

And  so  on  to 

sin.  14'  3"  45'"  =  .004090604. 

etc.  etc.  etc. 

The  following  elegant  method  of  deducing,  from  the 
sine  of  an  arc,  the  sine  and  cosine  of  one  half  the  arc,  is 
given,  assuming  that  the  student  is  familiar  with  the 
simple  algebraic  principles  upon  which  it  depends. 

Let  us  take  the  natural  sine  of  18°,  which  is  .3090170, 

18° 
and  make  x  =  sine,  and  y  the  cosine  of  9°  =  -— -. 

Then,  a:*  +  2/'  =  1;  W 

and  2xy  =  .3090170     (2);    Eq.  (30) 

Adding,  we  have 

jr'+  2xy  +/  =  1.3090170; 


SECTION    1.  269 

Taking  the  square  root,  we  have 

a:  +  3/  =   1.144123.     (3) 
Subtracting  (2)  from  (1), 

x^  —  2xy  +  y'  =  .690983; 
taking  the  square  root, 

rr  —  y  =  —.831254*         (4) 
Adding  (3)  and  (4),  2x  =    .312869, 

hence,  x  =  8in.9°  =    .1564345 

Subtracting  (4)  from  (3),       2^^  =  1.975377 
hence,  y  =  cos.9°   =    .9876885 

Now,  by  making  x  =  the  sine  of  4°  30',  and  y  =  cosine 
of  4°  30',  and  as  before 

x»  +  i/»  ==  1 
and  2xy  =  .1564345, 

we  obtain  the  sine  and  cosine  of  4°  30';  and  another  ope- 
ration will  give  the  sine  and  cosine  2°  15',  etc.,  etc. 

We  may  in  this  manner  compute  the  sines  and  cosines 
of  all  arcs  resulting  from  the  division  of  18°  by  2,  and 
we  may  make  their  values  accurate  to  any  assigned  deci- 
mal figure. 

This  has  been  carried  far  enough  to  show  how  a  table 
of  natural  sines,  etc.,  could  be  computed ;  but  in  conse- 
quence of  the  tedious  numerical  operations  which  the 
process  requires,  other  methods  are  resorted  to  in  the 
actual  construction  of  the  table. 

The  Calculus  furnishes  formulae  giving  the  values  of 
the  sines  and  cosines  of  arcs  developed  into  rapidly  con- 
verging series,  and  from  these  the  sines  and  cosines  of 
all  arcs  from  0°  to  90°,  can  be  determined  with  great 

*  AVhen  an  arc  is  less  than  45°,  the  cosine  exceeds  the  sine;  and 

when  the  arc  is  between  45°  and  90°,  the  sine   exceeds  the  cosine. 

Hence,  when  the  arc  is  9°,  y,  its  cosine,  exceeds  x,  its  sine ;  and  we 

therefore  plac  »,d  tJie  minus  sign  before  the  second  member  of  Eq.  (4). 

23* 


270 


PLAJE    TRIGONOMETRY. 


accuracy  and  with  comparatively  little  labor.  In  the  lajjt 
two  columns  on  each  page  of  Table  II,  will  be  found  the 
v^alues  thus  computed  of  the  sines  and  cosines  of  every 
degree  and  minute  of  a  quadrant. 

TRTGONOMETRICAL  LINES  FOR  ARCS  EXCEEDING  90°. 


From  the  annexed  figure, 
the  construction  of  which 
needs  no  explanation,  are 
deduced  by  simple  inspec-  ^ 
tion  the  results  given  in  the 
following 


TABLE. 


90°  +  a° 

270°  — a° 

Bin.  =   COS.  a,  COS.  =  —  sm.  a 

sin.  =  —  cos.  a,  cos.  =  —  sin.  a 

tan .  =  —  cot.  o,  cot.  =  —  tan .  a 

tan.  =   cot.  a,  cot.  =   tan.  a 

sec.  =  —  cosec.  a,  cosec.  =  sec.  a 

sec.  —  —  cosec.  a,  cosec.  =  —  sec.  a 

180°  — o° 

270°  4- a° 

sin.  =   shi.  a,  cos.  =  —  cos.  a 

sm.  =  —  cos.  o,  COS.  =   sm.  a 

tan.  =  —  tan.  a,  cot.  =  —  cot.  a 

tan.  =  —  cot.  a,  cot.  =  —  tan.  a 

sec.  =  —  sec.  a,  cosec.  =  cosec.  a 

sec.  =   cosec.  a,  cosec.  =  —  sec.  a 

180° -fa° 

360° -a° 

sin.=— sin.  a,cos.  =— cos.  a 

sin..  =  —  sin.  a,  cos.  =  cos.  a 

tan.=  tan.a,cot.  =  cot.  a 

tan.  =  —  tan.  o,  cot.  =  —  cot.  a 

8ec.  =— sec.  a,  co8ec.=— cosec.a 

sec.  =   sec.  a,  cosec.  = — cosec.  a  | 

1 

By  means  of  this  table,  the  values  of  the  tngonoraet- 
rical  lines  of  any  arc  between  90°  and  360°,  can  be  ex- 
pressed by  those  of  arcs  less  than  90°. 

K,  for  examj)le,  the  arc  is  118°,  we  have 


SECTION    1.  271 

Sin.  118°  =  sin.  (90°  +  28°)  =       cos.28°  ; 
tan. 118°  =  tau.(90°  -f  28°)  =  —  cot.28° ; 
etc.,  etc.,  etc. 

For  the  arc  230°,  we  have 

sin.  230°  =  sin.  (270°  —  40°)  =  —  cos     40°  ; 
flec.230°  =  6ec.(270°  —  40°)  =  —  cosec.40° ; 
etc.,  etc.,  etc. 

In  many  investigations,  it  becomes  necessary  to  con- 
M  iQv  the  functions  of  arcs  greater  than  360°  ;  but  since 
the  addition  of  360°  any  number  of  times  to  the  arc  a, 
will  give  an  arc  terminating  in  the  extremity  of  a,  it  id 
obvious  that  the  arc  resulting  from  such  addition  will 
have  the  same  functions  as  the  arc  a.  And  hence  it  fol- 
lows that  the  functions  of  arcs,  however  great,  may  be 
ftxpressed  in  terms  of  the  functions  of  arcs  less  than  90°. 


272 


PLANE    TRIGONOMETRY. 


SECTION    II. 


PLANE   TRIGONOMETRY,  PRACTICALLY  APPLIED. 

In  the  preceding  section,  the  theory  of  Trigonometry 
has  been  quite  fully  developed,  and  the  student  should 
now  be  prepared  for  its  various  applications,  were  he 
acquainted  with  logarithms.  But  logarithms  are  no  part 
of  Trigonometry,  and  serve  only  to  facilitate  the  numeri- 
cal operations.  Trigonometrical  computations  can  be 
made  without  logarithms,  and  were  so  made  long  before 
the  theory  of  logarithms  was  understood. 

For  this  reason,  we  proceed  at  once  to  the  solution  of 
the  following  triangles. 

1.  The  hypotenuse  of  a  right-angled  triangle  is  21, 
and  the  base  is  17 ;  required  the  perpendicular  and  the 
acute  angles. 

Let  CAB  be  the  triangle,  in 
which   OB  =  21,  and    CA  z=         ■  p 

17.  With  C  as  a  center,  and 
CD  =  1  as  a  radius,  describe 
the  arc  DE,  of  which  the  sine 
IS  DF^  the  tangent  is  EG,  and 
the  cosine  is  CF. 

By  similar  triangles  we  have 
CB  :  CA  '. 

that  is,  21  :  17     : 

17 
21 


Hence, 


COS.  C 


I 

C            .           1 

FE             A 

CD  :   CF', 

1       :  cos.  C. 

=  .80952+. 

SECTION    II.  27a 

We  must  now  turn  to  Table  II,  and  find  in  the  last  two  columns 
the  cosine  nearest  to  .80952,  and  the  corresponding  degrees  and 
minutes  will  be  the  value  of  the  angle  C. 

On  page  57,  of  Tables,  near  the  bottom  of  the  page,  and  in  the 
column  with  cosine  at  the  top,  we  find  .80953,  which  coriesponda 
to  35*"  56'  for  the  angle  C.     The  angle  B  is,  therefore,  54°  3'. 

'T'ais  Table  is  so  arranged,  that  the  sum  of  the  degrees  at  the  top 
and  bottom  of  the  page,  added  to  the  sum  of  the  minutes  which  are 
found  on  the  same  horizontal  line  in  the  two  side  columns  of  the 
pagC;  is  90° 

Thus,  in  finding  the  angle  C,  the  number  .80953  was  found  in 
the  column  with  cosine  at  the  head.  /  We  therefore  took  the  de« 
grees  from  the  head  of  the  page,  and  the  minutes  were  taken 
from  the  left  hand  column,  counting  downwards. 

For  the  side  ABy  we  have  the  proportion 

CF  :  FB  ::   CA  :  AB] 

or,  cos.  0  :  sin.  C  i:  17  :  AB] 

that  is,  .80953  :  .58708  : :  17  :  AB. 

From  which  we  find  AB  =  .58708  x  17  -i-  .80953; 

whence,  AB  =  12.328. 

If  we  had  formed  a  table  of  natural  tangents,  as  well  as  of  natii- 
ral  sines,  AB  could  have  been  found  by  the  following  proportion  • 
CE  :  EG  ::  CA  :  AB 

or,  1  :  tan.  C  ::  17  :  AB; 

whence,  AB  =  17  tan.  C. 

The  perpendicular  AB  may  also  be  found  b}'  the  proportion 
CD  :  DF  ::   CB  :  AB; 

or,  1  :  sin.  C  ::  21  :  AB; 

whence,    AB  =  21  sin.  C  =  21  x  .58708  =  12.32868. 

2.  The  two  sides  of  a  right-angled  triangle  are  150  ?nd 
125 ;  required  the  hypotenuse  and  the  acute  angles. 

We   may  employ  the   same 
figure  as  in  the  preceding  prob-  ^ 

lem. 

Then,  from  the  similar  trian- 
gles, CFD  and  CAB,  we  get 

CF  :  FD  ::  CA  :  AB; 


274  PLANE    TRIG  ONOMETBY. 

that  is,        COS.  C  :  sin.  C  : :  150  :  125  : :  6  :  6, 
which  gives  6  sin.  6^  =   5  cos.  0} 

hence,  36  sin.'C   =  25  cos.«C. 

A.dding  member  to  member,     36  cos.^6'  =  36  cos.*C. 

we  have  36  (sin.^C4- cos.'^C^  =   61  cos.* C. 

Butsin.^C-f  cos.'C  =  1,  (Eq.  (1)  Trigonometry); 
whence,  61  cos.^C7  =  36; 

cos.^a  ==  '^  =   .5901639344: 
61 

and  COS.  C  =  .76822,  nearly. 

To  find  the  angle  of  which  this  is  the  cosine,  we  turn  to  page  6(1 
of  tables,  and  looking  in  the  column  having  cosine  at  the  head;  we 
Bee  that  .76822  falls  between  .76828,  which  has  48'  opposite  to  it 
in  the  left  hand  column,  and  .76810,  which  has  49'  opposite  to  it 
in  the  same  column.  Now,  the  cosines  of  arcs  less  than  90°  de- 
crease when  the  arcs  increase,  and  the  converse ;  and  while  the 
increase  of  the  arc  is  confined  within  the  limits  of  1',  the  increase 
of  the  arc  will  be  sensibly  proportional  to  the  decrease  of  th^  cosine. 

0.76828  .76828 

Hence,  0.76810  .76822 

18       :  6  : :  60"  :  sf' 

which  gives  a;"  =  20". 

The  angle  C  is,  therefore,  equal  to  39°  48'  20",  and  the  angle 
B  s  90°  —  39°  48'  20"  =  50°  11'  40". 
To  find  CB,  we  have 

CF  :  (W  : :   CA  i  CB 

or,  COS.C  :  1  ::  150  :   CB 

that  is,  .76822  :  1  ::  150  :   CB 

150 
whence,  CB  =  --4>o  =   1^5.26—. 

3.  The  base  of  a  right-angled  triangle  is  150,  and  the 
tii  gle  opposite  the  base  is  50°  11'  40"  ;  required  the 
h>  potenuse  and  the  perpendicular. 


oECTION    II. 


275 


Let  CAB  be  the  triangle. 
Then,  (Prop.  4,  Sec.  I), 
riu.50°  11'  40"  ;  sin.  90"  ::  150  :  CB. 
Whence, 


CB 


150 


:j;j  =  195.26, 


.708:^2 

»lie  same  as  in  the  preceding  example. 
To  find  ABj  we  have 

CD  :  DF  ::  CB  :  AB) 
that  is,  1  :  sin.  (7  or  cos.  5  ::  195.26  :  AB) 

from  which  we  find 

AB  =  195.26  sin.  39°  48' 40"; 
or,  AB  =^  125.0015. 

4.  Two  sides,  the  one  30  and  the  other  35,  and  the  in- 
cluded angle  20°,  of  a  triangle,  are  given,  to  find  the 
other  two  angles  and  the  third  side. 

*Let  BA  C  be  the  triangle,  in  which  BC 
=  35,  BA  =  30,  and  the  angle  B  = 
20°.  From  A^  the  extremity  of  the 
shorter -side,  let  fall  on  BC  the  perpen- 
dicular AD,  thus  dividing  the  triangle  ^ 
into  the  two  right-angled  triangles  BAD  and  CAD. 

Then,  from  the  triangle  BAD,  we  have 


1st, 

or, 

2d. 


BA 

:  AD) 

30 

'.  AD  ^  m  sin.  20** 

BA 

:BD; 

30 

:  BD  =  BO  cos.  20° 

sin.  D  :  sin.  B 
1  :  sin.  20° 
1  :  COS.  B 
or,  1  :  COS.  20° 

In  the  table  of  natural  sines,  we  find  sin.  20°  =  .34202,  and  tht 
.'OS.  20°  =x  .93969;  hence,  AD  =  30  X  .34202  =  10.26060,  and 
fii;  =  30  X  .93969  =  28.19070,  and  therefore  DC  :=  BC -^ 
BD  =  6.8093. 
From  the  triangle  CAD,  we  have 


1st,  AC=^  ^AD'  +  DC^  =  v/(10.26)*  +  (6.8+/  ^  '»*'  ^^'^ 
2d.  Ar^  :  AD  ::  Sxn.90°  :  sin.  C: 


276 

or, 

whence, 
and  the 


PLANE    TRIGONOMETRY. 
12.367  :  10.26+  ::  1  ;  sin.  C) 

«^"-  ^  =  1W7  =  -^^^l^- 
ande   C  =  56°  26\ 


If,  now,  we  add  angles  B  and  (7,  and  take  the  sum  from  180* 
the  remainder  will  be  the  angle  BA  G, 

Hence,     \__BAC=^  180°  —  (56° 26'+  20°)  =  103°  34'. 

5.  Two  sides,  the  one  18  and  the  other  21,  and  the 
angle  opposite  the  side  24  equal  to  76°,  are  given,  to  find 
the  remaining  side  and  the  other  two  angles. 
Let  X  denote  the  angle  opposite  the  side  18.     Then, 

24  :  18  ::  sin.  76°  :  sin.ar,  (Prop.  4,  Trig.). 
or,  4:3::  sin.  76°  :  sin.  x. 

sin.  a;  =  f  sin.  76°  =  f  X  .97030  =  .72772; 

whence  the  angle  opposite  the  side  18  is  46°  41'  45". 
Adding  this  to  the  given  angle,  and  taking  the  sum  from  18V®, 
we  get  57°  18'  15"  for  the  third  angle. 

To  find  the  remaining  side,  denoted  by  y,  we  have 

sin.  76°  :  sin.  57°  18'  15"  ::  24  :  y; 

or,  .97030  :  .84155  : :  24  :  y. 

24  X  .84155 


.97030 


=  20.815  =  3d  side. 


6.  The  three  sides  of  a  triangle  are  18,  24,  and  20.816  j 
required  the  angles. 

This  problem  may  be  solved  by  Prop.  6,  or  by  Prop.  8,  Trigo- 
nometry. 

First     By  Prop.  6. 

In  the  triangle  ABCy  make  CB  = 
24,  ^(7=  20.815,  and  AB  =  18. 

Then, 

24  :  38  815  ::  2.815  :   CD -^  BD. 

e7>-5i>=  1^-264225^ 
24 


SECTION    II.  277 

Put  CD  +  BD  ^    OB  =  21. 

By  addition,  we  get       2CD  ^  28.5527; 

dividing  by  2,  and  CD  =   14.2763+. 

And  hence,  BD  =   CB  —  CD  =  24  —  14.2763  =  9.7237. 

In  the  triangle  ADB,  we  have 

BA  :     BD     ::  1  :  cos.  5 

or,  18  :  9.7237  ::  1  :  cos.  ^  =  .54020 

rr  ki    TT   T>       ^Q     (cos.  57°  18'  =  .54024) 
Table  11,  Page  53,   |  ,,,  5,0  .p.  ^  .^^ppp  } 

diff.  =  24  :  60"  : :  4  :  10" 

hence,  L  ^  =  ^7*^  18'  10". 

It  will  be  observed  that  Examples  5  and  6  refer  to  the  same  tri- 
ungle,  and  that  in  Example  5  the  angle  B  was  57°  18'  15".  Thia 
slight  discrepancy  in  the  results  should  be  expected,  on  account  of 
the  small  number  of  decimal  places  used  in  the  computations. 

Second.     By  Prop.  8. 

Sum  of  the  sides,                               =  62.815, 
half  sura  denoted  by  s,                       =  31.4075 
a  =  24 

s  —  a  =    7.4075 


Formula,  cos.  \  A  =  \  /  —^- ^,  radius  being  unity. 

s{s  —  d)=z  31.4075  X  7.4075  =  232.65105625 
he  =  20.815    X         18  =  374.67 
^^^"""-^  «   .62095  very  nearly. 

V':62095  =  .78800. 

Hence,  cos.  \A  =  .78800,  and  M  (Table  II,  page  59)  »  SS* 
Tery  nearly ;  the  angle  A  is  therefore  equal  to  76°,  which  agrees 
with  Example  5. 

7.  Given,  the  three  sides,  1425, 1338,  and  493,  of  a  tri- 
angle; required,  the  angle  opposite  the  greater  .side,  using 
the  formula  for  the  sine  of  one  half  an  angle. 
24 


278  PLANE    TRIGONOMETRY. 

Make  a  =  1425,  h  x=  1338,  and  c  =  493 ;  then  the  [_  i  ii 
opposite  the  side  a,  and  the  formula  is 

Ann  A  =  (^  -  ^)  («  -A 
be 

in  which  s  denotes  the  half  sum  of  the  three  sides. 

Then  we  have  s  =  1628,  s  —  h  =  290,  s^c=^  1135,  (s  —  h) 

is-^c)  =  329150,  ic  =  659634,  (^  — ^)  (^  — 0  ^  .498988 

be 


H.nce,  sin.  U  =  ^.498988  =  .70632. 

In  the  table  we  find         sin.  44°  5Q'  28.5"  ~  .70638. 
Therefore,  lA  =  44°  56'  28.5",  and  A  =  89°  52'  5l"  ;  —  but 
little  less  than  a  right  angle. 

In  these  seven  examples  we  have  shown  that  it  is  possi- 
ble to  solve  any  plane  triangle,  in  which  three  parts,  one 
at  least  being  a  side,  are  given,  without  the  aid  of  loga- 
rithms. But,  when  great  accuracy  is  required,  and  the 
number  of  decimal  places  employed  is  large,  the  necessary 
multiplications  and  divisions,  the  raising  to  powers,  and 
the  extraction  of  roots,  become  very  tedious.  All  of  these 
opera tiors  may  be  performed  without  impairing  the  cor- 
rectness of  results,  and  with  a  great  saving  of  labor,  by 
means  of  logarithms ;  but,  before  using  them,  the  student 
should  be  made  acquainted  with  their  nature  and  pro- 
perties. 

LOGARITHMS.      ^ 

Logarithms  are  the  exponents  of  the  powers  to  which 
fi  fixed  number,  called  the  base,  must  be  raised,  to  pro- 
duce other  numbers. 

The  exponent  of  a  number  is  also  a  number  express- 
ing how  many  times  the  first  number  is  taken  as  a  factor. 

Thus,  let  a  denote  any  number ;  then  a'  indicates  that  a 
has  been  used  three  times  as  a  factor,  a*  that  it  has  been 
aoe.l  four  times  as  a  factor,  and  a"  that  it  has  been  thus 
•i3cd  n  times. 


OECTION    II.  279 

Kow,  instead  of  calling  these  numbers  S,  4, n, 

exponents,  we  call  them  the  logarithms  of  the  powers  a », 
aS a\ 

To  multiply  a^  by  a',  we  have  simply  to  ^^Tite  a,  giving 
it  an  exponent  equal  to  2  +  5 ;  thus,  a^  x  a^  —  a\ 

Hence,  the  sum  of  the  logarithms  of  any  number  of  factoit 
is  equal  to  the  logarithm  of  the  product. 

To  divide  a"  by  a»,  we  have  oul}^  to  write  a,  giving  it 
au  exponent  equal  to  12  —  9;  thus,  a"-r-a'  =  a';  and, 
gererally,  the  quotient  arising  from  the  division  of  a"*  by 
a",  is  equal  to  a"'~". 

Hence,  the  logarithm  of  a  quotient  is  the  logarithm  of  the 
dividend  diminished  hy  the  logarithm  of  the  divisor. 

If  it  is  required  to  raise  a  number  denoted  by  a',  to  the 
fifth  power,  we  write  a,  giving  it  an  exponent  equal  to 
3x5;  thus,  {a^)^=  a ",  and,  generally,  (a «) "» =  a " *". 

Hence,  the  logarithm  of  the  power  of  a  number  is  equal  to 
the  logarithm  of  the  number  multiplied  by  the  expoiient  of  the 
power. 

To  extract  the  5th  root  of  the  number  a',  we  wTite  a, 
giving  it  an  exponent  equal  to  |;  thus,  \/a'=a^,  and, 
generally,  to  extract  any  root  of  a  number,  we  divide  the 
exponent  of  the  number  by  the  index  of  the  root,  and  the 
quotient  will  be  the  exponent  of  the  required  root. 

Hence,  the  logarithm  of  a  root  of  a  number  is  equal  to  the 
quotient  obtained  by  dividing  the  logarithm  of  the  number  by 
the  index  of  the  root. 

Now,  understanding  that  by  means  of  a  table  of  loga- 
rithms we  may  find  the  numbers  answering  to  given 
logarithms,  with  as  much  facility  as  we  can  find  the  loga- 
rithms of  giv^en  numbers,  we  see  from  what  precedes  that 
multiplications,  divisions,  raising  to  powers,  and  the  ex- 
traction of  roots,  may  be  perfonned  by  logarithms ;  and 
the  utility  of  logarithms,  in  trigonometrical  computations, 
mainly  consists  in  the  simplification  and  abridgment  of 
these  operations  by  their  use. 


280  PLANE    TRIGONOMETRY. 

The  common  logarithms  are  those  of  which  10  is  the 
base ;  that  is,  they  are  the  exponents  of  10. 

Thus,  10'  =  10  Hence  the  logarithm  10       =1. 

10^=100  "        "            "         100      =2. 

10' =1000  "        <             "         1000    =3. 

10*  =  10000  "        "           «        10000  =  4. 

etc.      etc.  etc.                   etc.     etc. 

10  dr 

Since  2Q  =  1  =  10'-'  =  10",  and  generally  -^^  =  a"  =  1, 

it  follovva  that  in  this,  as  in  all  other  systems,  the  loga- 
rithm ot  1  =  0. 

From  what  precedes,  it  is  evident  that  the  logarithm  of 
any  number  between  10  and  100  must  be  found  between 

1  and  2 ;  that  is,  its  logarithm  is  1  plus  a  number  less 
than  1;  and  any  number  between  100  and  1000,  will 
have  for  its  logarithm  2  plus  some  number  less  than  1, 
and  so  on.  The  fractional  part  of  the  logarithm  of  a 
number  is  expressed  decimally. 

The  entire  number  belonging  to  a  logarithm  is  called 
its  index.  The  index  is  never  put  in  the  tables,  (except 
from  1  to  100),  and  need  not  be  put  there,  because  we 
always  know  what  it  is.  It  is  always  one  less  than  the 
number  of  digits  in  the  integer.  Thus,  the  number  3754 
has  3  for  the  index  to  its  logarithm,  because  the  number 
consists  of  4  digits;  that  is,  the  logarithm  is  3  and  some 
decimal. 

The  number  347.921  has  2  for  the*  index  of  its  loga- 
rithm, because  the  number  is  between  347  and  348,  and 

2  is  the  index  for  the  logarithms  of  all  numbers  over  100, 
and  less  than  1000. 

All  numbers  consisting  of  the  same  figures,  whether 
integral,  fractional,  or  mixed,  have  logarithms  consisting 
of  the  same  decimal  part.     The  logarithms  differ  only  in 
their  indices. 
24* 


SECTION    II. 


281 


Thus, 


the  number 
the  number 
the  number 
the  number 
the  number 


7956.  has 
795.6  has 
79.56  has 
7.956  has 


3.900695  for  its  log. 
2.900695 
1.900695 
0.900695 


.7956  has  —1.900695 


the  number  .07956  has  —2.900695 

From  this  we  perceive  that  we  must  take  the  logarithm 
01' t  of  the  table  for  a  mixed  number  or  a  decimal,  the 
same  as  if  the  figures  expressed  an  entire  number;  and 
then,  to  prefix  the  index,  we  must  consider  the  value  of 
the  number. 

The  decimal  part  of  a  logarithm  is  always  positive; 
Dut  the  index  becomes  negative  when  the  number  is  a 
decimal ;  and  the  smaller  the  decimal,  the  greater  the 
negative  index.     Hence, 

To  prefix  the  index  to  a  decimal,  count  the  decimal 
point  as  1,  and  every  cipher  as  1,  up  to  the  first  significant 
figure,  and  this  is  the  negative  index. 

For  example,  find  the  logarithm  of  the  decimal 
.0000831. 

ISTum.  .0000831;  log.  —5.919601. 

The  point  is  counted  one,  and  each  of  the  ciphers  is 
counted  one;  therefore  the  index  is  minus  five. 

The  smaller  the  decimal,  the  greater  the  negative 
index ;  and  when  the  number  becomes  0,  the  logarithm  is 
negatively  infinite. 

Hence,  the  logarithmic  sine  of  0°  is  negativehj  infimite^ 
liowever  great  the  radius. 

A  number  being  given,  to  find  its  corresponding  logarithm. 

The  logarithn  of  any  number  consisting  of  four  figures, 
or  less,  is  taken  out  of  the  table  directly,  and  without  the 
least  difficulty. 

Thus,  to  find  the  logarithm  of  the  number  3725,  wf 
24  * 


282  PLANE    TRlGONOxME  TRY 

find  372  at  the  side  of  the  table,  and  in  the  columfl 
marked  5  at  the  top,  and  opposite  372,  we  find  .571126, 
for  the  decimal  part  of  the  logarithm. 

Hence,  the  logarithm  of  3725  is  3.571126. 
the  logarithm  of  37250  is  4.571126. 
the  logarithm  of  37.25    is  1.571126,  etc. 

Find  the  logarithm  of  the  number  834785. 
This  number  is  so  large  that  we  cannot  find  it  in  the 
table,  but  we  can  find  the  numbers  8347  and  8348.  The 
logarithms  of  these  numbers  are  the  same  as  the  loga- 
rithms of  the  numbers  834700  and  834800,  except  the 
indices. 

834700    log.    5.921530 
834800    log.    5.921582 


DiiFerence,  100  52 

Now,  our  proposed  number,  834785,  is  between  the 
two  assumed  numbers ;  and,  of  course,  its  logarithm  lies 
between  the  logarithms  of  the  two  assumed  numbers; 
and,  without  further  connnent,  we  may  find  it  by  propor- 
tion thus, 

'100  :   85  =  52  :  44.2 

Or,  1.  :  .85  =  52  :  44.2 

Hence,  for  finding  from  the  table  the  logarithm  ot  a 
number  consisting  of  more  than  four  places  of  figures, 
we  have  the  following 

RULE. 

Take  from  the  table  the  log.  of  the  number  expressed  by  the 
the  four  superior  figures  ;  this,  with  the  proper  index,  is  the 
approximate  logarithm.  Multiply  the  number  expressed  by  the 
remaining  figures  of  the  number,  regarded  as  a  decimal,  by 
the  tabular  difference,  and  the  product  will  be  the  correction 
to  he  added  to  the  approximate  log.  to  obtain  the  true  log 


SECTION    II.  283 


EXAMPLES. 


1.  What  is  the  log.  of  357.32514? 

The  log.  of  357.3  is  2.553033 

No.  not  included,     .2514 
Tabular  diflf.,  122 

Prod.,  30.6708 ;  correction,    81 

log.  sought,  2.553064 

The  log.  of  35732.514  is  4.553064 

"  .035732514  "         —2.553064. 

2.  What  IS  the  log.  of  7912532  ? 

Approximate  log.,  6.898286 
.532  X  55  =   correction,  29 


True  log.  =  6.898315. 

A  logarithm  being  given,  to  find  its  corresponding  number. 

For  example,  what  number  corresponds  to  the  log. 
6.898315? 

The  index  6  shows  that  the  entire  part  of  the  number  must  con« 
tain  seven  places  of  figures.  With  the  decimal  part,  .898315,  of 
the  log.,  we  turn  to  the  table,  and  find  the  next  less  decimal  part 
to  be  .898286,  which  corresponds  to  the  superior  places,  7912. 

The  difiercnce  between  the  given  log.  and  the  one  next  less  is 
29.  This  we  divide  by  the  tabular  difference,  55,  because  we  are 
working  the  converse  of  the  preceding  problem.     Thus, 

29  H-  55  =  52727-f . 

Place  the  quotient  to  the  right  of  the  four  figures  before  found, 
And  we  shall  have  7912527.27  for  the  number  sought. 

This  example  W3.S  taken  from  the  preceding  case,  and 
the  number  found  should  have  been  7912532 ;  and  so  it 
would  have  been,  had  we  used  the  true  difterence,  29.26, 
in  place  of  29. 

When  the  numbers  are  larere,  as  in  this  example,  the 


284  PLANE    TRIGONOMETRY. 

result  is  liable  to  a  small  error,  to  avoid  which  the  loga- 
rithms should  contain  a  great  number  of  decimal  places; 
but  the  logarithms  in  our  table  contain  a  sufficient  num- 
ber of  decimal  places  for  most  practical  purposes. 

Hence,  for  finding  the  number  corresponding  to  a  ay 
given  logarithm,  we  have  the  following 

RULE. 

Look  in  the  table  for  the  decimal  part,  of  the  given  lego- 
rithm,  and  if  not  found,  take  the  decimal  next  less,  and  take 
out  the  four  corresponding  figures. 

Take  the  difference  between  the  given  log.  and  the  next  less 
in  the  table  ;  divide  that  difference  by  the  tabular  difference, 
and  write  the  quotient  on  the  right  of  the  four  superior  fig^ 
ures,  and  the  result  is  the  number  sought. 

Point  off  the  whole  number  required  by  the  given  index, 

EXAMPLES. 

1.  Given,  the  logarithm  3.743210,  to  find  its  corres- 
ponding number  true  to  three  places  of  decimals. 

Ans.  5536.177. 

2.  Given,  the  logarithm  2.633356,  to  find  its  corres- 
ponding number  true  to  tw^o  places  of  decimals. 

\,_^S^^  ^ns.  429.89. 

3.  Given,  the  logarithm  — 3.291746,  to  find  its  corres- 
ponding number.  Ans.  .0019577. 

4.  What  number  corresponds  to  the  log.  3.233568  ? 

Ans.  1712.25. 
\  What  is  the  number  of  which  1.532708  is  the  log.  T 

Ans.  34.0963. 
6.  Find  the  number  whose  log.  is  1.067889. 

Ans.  11.692. 

EXPLANATION    OF    TABLE    II. 

Table  I  is  merely  a  table  of  numbers  and  their  corres- 
ponding logarithms,  and  requires  no  explanation  other 


SECTION    II.  286 

than  that  which  has  been  given  in  connection  with  the 
suljject  of  loganthms. 

Table  II,  with  the  exception  of  the  last  two  oohmnib, 
which  contain  natural  sines  and  cosines,  is  a  table  in 
which  are  arranged  the  logarithms  of  the  numerical 
valucc  of  the  several  trigonometrical  lines  corresponding 
to  the  different  angles  in  a  quadrant.  The  values  of 
these  'ines  are  computed  to  the  radius  10,000,000,000, 
and  i^eir  logarithms  are  nothing  more  than  the  loga- 
rithms, each  increased  by  10,  of  the  natural  sines,  co- 
sines, and  tangents,  of  the  same  angles ;  because  the 
values  of  these  lines,  for  arcs  of  the  same  number  of  de- 
grees tiiken  in  different  circles,  are  directly  proportional 
to  the  radii  of  the  circles. 

The  natural  sines  are  made  to  the  radius  of  unity; 
and,  of  course,  any  particular  sine  is  a  decimal  fraction, 
expressed  by  natural  numbers.  The  logarithm  of  any 
natural  sine,  with  its  index  increased  by  10,  will  give 
the  logarithmic  sine.  Thus,  the  natural  sine  of  3°  is 
.052336. 

The  logarithm  of  this  decimal  is  —  2.718800 

To  which  add  10. 

The  logarithmic  sine  of  3°  is,  therefore,  8.718800 

In  this  manner  we  may  ffnd  the  logarithmic  sine  of 
any  other  arc,  when  we  have  the  natural  sine  of  the 
same  arc. 

If  the  natural  sines  and  logarithmic  sines  were  on  the 
eame  radius,  the  logarithm  of  the  natural  sine  would  be 
the  logarithmic  sine,  at  once,  without  any  increase  of 
the  index. 

The  radius  for  the  logarithmic  sines  is  arbitrarily 
tiiken  so  large  that  the  index  of  its  logarithm  is  10.  It 
might  have  been  more  or  less ;  but,  by  common  consent, 
it  is  settled  at  this  value;  so  that  the  sines  of  the  smallest 
area  over  used  shall  not  have  a  negative  index. 


286  PLANE    TRIGONOMETRY 

In  our  preceding  equations,  sin.  a,  cos.  a,  etc.,  rofei 
to  natural  sines;  and  by  such  equations  we  determine 
their  values  in  natural  numbers;  and  these  numbers  are 
put  in  Table  II,  under  the  heads  of  iT.  sine  and  M,  cos., 
as  before  observed. 

When  we  have  the  sine  and  cosine  of  an  aye,  the 
tangent  and  cotangent  are  found  by  Eq.  (3)  and  (6) ;  thus, 

,  E  sin.     ,^.       ,        E  COS. 

tan.  =  ( o )  cot.  =-  -  -  — : 

COS.  6.i»:. 

and  the  secant  is  found  by  equation  (-^'/j  that  is, 

E' 

sec.  = 

COS. 

For  example,  the  logarithmic  sine  of  6°  is  9.01923^, 
and  its  cosine  9.997614.  From  these  it  is  required  to 
find  the  logarithmic  tangent,  cotangent,  and  secant. 

R  sin.  19.019235 

Cos.  subtract    9.997614 


Tan.  is 

9.021621 

E  cos. 

19.997614 

Sin. 

subtract    9.019235 

Cotan.  is 

10.978379 

R'is 

20.000000 

Cos. 

subtract     9.997674 

Secant  is  10.002326 

The  secants  and  cosecants  of  arcs  are  not  given  in 
our  table,  because  they  are  very  little  used  in  practice ; 
and  if  any  particular  secant  is  required,  it  can  be  deter- 
mined by  subtracting  the  cosine  from  20;  and  the  cose- 
cant can  be  found  by  subtracting  the  sine  from  20. 

The  sine  of  every  degree  and  minute  of  the  quadrant 
is  given,  directly,  in  the  table,  commencing  at  0°,  and 
extending  to  45°,  at  the  head  of  the  table ;  and  from  45° 
to  90°,  at  the  bottom  of  the  table,  increasing  backward. 


SECTION    II.  2S1 

Tho  column  having  sine  at  tlie  top  has  cosine  at  the 
lK)ttom,  and  the  opposite,  because  angles  read  from  abo^  e 
are  complementary  to  those  read  from  below.  The  differ- 
ences of  consecutive  logarithms  corresponding  to  10"  are 
given  for  both  sine  and  cosine,  but  the  tangents  and  cotan- 
gents have  the  same  column  of  differences  for  the  reason 
that  log.  tan. -flog.  cot.=log.  R^  and  is  therefore  constant, 
llence,  by  just  as  much  as  log.  tan.  increases,  log.  cot.  de- 
sreases  and  the  converse. 

Ab  cosines  and  cotangents  decrease  when  arcs  increase, 
and  increase  when  arcs  decrease,  the  proportional  parts 
answering  to  seconds  for  them  must  be  subtracted. 

Mca?n2?le.  Find  the  sine  of  19°  17'  22". 

The  sine  of  IQ**  17',  taken  directly  from  the  table,  is  9.518829 

The  difference  for  10"  is  60.2;  for  1"  is  6.02;  and 

for  6.02  X  22  =  132 

flence,  log.  sine  19"  17'  22"  is  9.518961 

From  this  it  will  be  perceived  that  there  is  no  difficulty 
in  obtaining  the  sine  or  tangent,  cosine  or  cotangent,  of 
any  angle  greater  than  30'. 

Conversely :  Given,  the  logarithmic  sine  9.982412,  to 
find  its  corresponding  arc.  The  sine  next  less  in  the 
table  is  9.982404,  which  gives  the  arc  73°  48'.  The  differ- 
ence between  this  and  the  given  sine  is  8,  and  the  dif- 
ference for  V  is  .61 ;  therefore,  the  number  of  seconds 
corresponding  to  8,  must  be  discovered  by  dividing  8  by 
the  decimal  .01,  which  gives  13.  Hence,  the  arc  sought 
LS  73°  48' 13". 

These  operations  are  too  obvnous  to  require  a  rule. 
SVTien  the  arc  is  very  small, — and  such  arcs  are  sometimes 
required  in  Astronomy, —  it  is  necessary  to  be  very  accu- 
rate; fortius  reason  we  omitted  the  difference  for  seconds 
or  all  arcs  under  30'.  Assuming  that  the  sines  and  tan- 
gents of  arcs  under  30'  vary  in  the  same  proportion  as 
tlie  arcs  themselves,  we  can  find  the  sine  or  tangent  of 
any  yqij  small  arc,  with  ^reat  exactness,  as  follows: 


288  PLANE    TRIGONOMETRY. 

The  sine  of  1',  as  expressed  in  the  table,  is  3.463726 

Divide  this  by  60;  that  is,  subtract  logarithm  1.778151 

The  logarithmic  sine  of  1",  therefore,  is  4.685575 

Now,  for  the  sine  of  17",  add  the  logarithm  of  17     1.230449. 

Logarithmic  sine  of  17",  is  5.916024 

In  the  same  manner  we  may  find  the  sine  of  any  other 
am  all  arc. 
For  example,  find  the  sine  of  14'  21  J";  that  is,  861.5". 

The  logarithmic  sine  of  1"  is  4.685575 

Add  logarithm  of  861.5,  2.935254 

Logarithmic  sine  of  14'  21  r',  7.620829 

Two  lines  drawn,  the  one  from  the  surface  and  the 
other  from  the  center  of  the  earth,  to  the  center  of  the 
sun,  make  with  each  other  an  angle  of  8.61".  What  ia 
the  logarithmic  sine  of  this  angle  ? 

The  log.  of  the  sine  1"  is  4.685575 

Log.  of  8.61,  0.935003 

Log.  sine  of  sun's  horizontal  parallax  =  5.620578 


GENERAL  APPLICATIONS  WITH  THE  USE  OF 
LOGARITHMS. 

L  EIGHT-ANGLED  TRIGONOMETRY. 

One  figure  will  be  sufiicient  to  represent  the  triangle 
lu  all  of  the  following  examples ;  the  right  ansjle  being 
at  B. 

PRACTICAL    PROBLEMS. 

1.  In  a  right-angled   triangle,  ABC,        ^ 
given  the  base  AB,  1214,  and  the  angle 
4,  51°  40'  30'',  to  find  the  other  parts. 


SECTION   II.  289 

To  fiud  BC. 

Radius,  10.000000 

:     tan.  ^,  51°  40' 30",  10.102119 

::    AB,  1214,  3.084219 

:     BC,  1535.8,  3.186338 

Remark. — AVhen  the  first  term  of  a  logarithmic  proportion  is  rad/UB, 
fche  required  logarithm  is  found  by  adding  the  second  and  third  loga- 
rithms, rejecting  10  in  the  index,  which  is  dividing  by  the  first  term. 

In  all  cases  we  add  the  second  and  third  logarithms  together ;  which, 
in  logarithms,  is  multiplying  these  terms  together ;  and  from  that  sum 
we  subtract  the  first  logarithm,  whatever  it  may  be,  which  is  dividing 
by  the  first  term. 

To  find  ^  a 

Sin.  (7,  or  cos.  A,  51°  40'  30",  9.792477 

:     AB,  1214,  3.084219 

.:    Radius,  10.000000 


:    AC,  1957.7,  3.291742 

To  find  this  resulting  logarithm,  we  subtracted  the  first  logarithm 
from  the  second,  conceiving  its  index  to  be  13. 

Let  ABO  represent  any  plane  triangle,  right-angled 
at  ^. 

2.  Given,  AC  73.26,  and  the  angle  A,  49°  12'  20"; 
required  the  other  parts. 

Ans.  The  angle  C,  40°  47'  40" ;  BC,  55.46 ;  and  AB,  47.86. 

3.  Given,  AB  469.84,  and  the  angle  A,  51°  26'  17",  to 
9  ad  the  other  parts. 

Ans.  The  angle  C,  38°  88' 43";'  BC,  588.7 ;  and  ^(7,  752.9. 

4.  Given,  AB  498,  and  the  angle  C,  20°  14' ;  required, 
the  remaining  parts. 

Ans.  The  angle  A,  69°  46';  BC,  1888 ;  and  AC,  1425.5. 

5.  Let  AB  =  831,  and  the  angle  A  =  49°  14' ;  what  are 
the  other  parts? 

Ans.  AC,  506.9;  BC,  388.9;  and  the  angle  C,  40°  46'. 

6.  If  ^(7=45,  and  the  angle  (7=87°  22',  what  are  the 
remaining  parts  ? 

Ans.  AB,  27.81;  5(7,  35.76;  and  the  angle  ^,  52°  38 
25  T 


290 


PLANE    TEIGOI^OMETRT. 


7.  Given,  ^(7=4264.3,  and  the  angle  J.  =  56°  29'  13", 
to  find  the  remaining  parts. 

Ans.AB,  2354.4  ;5C,  3555.4;  and  the  angle  (7, 33°  30'47". 

8.  If  4^  =  42.2,  and  the  angle  A  =  31°  12' 49'',  what 
are  the  other  parts  ? 

Ans,  A  0, 49.34 ;  BO,  25.57 ;  and  the  angle  0, 58°  47'  11". 
•j.  It  AB  =  8372.1,  and  BO  ^  694.73,  what  are  the 
:>ther  pai-ts? 

iAC,  8400.9  ;  the  angle  C,  85''  15'  23"  ;  and  the 
^''^^-    }  angle  A,  4^  44'  37". 

10.  If  AB  be  63.4,  and  AC  he  85.72,  what  are  the 
other  parts  ? 

i  BO,  57.69  ;   the  angle  C,  47°  41'  56' ;   and  the 
^^'^-  (      angle  ^,  42°  18'  4". 

11.  Given,  AQ  ==  7269,  and  ^^  =  3162,  to  find  the 
other  parts. 

.        f  ^(7,  6545 ;   the  angle  0,  25°  47'  7" ;   and  the 
'^^'  I      angle  ^,  64°  12'  53". 

12.  Given,  AO  =^  4824,  and  BO  r=  2412,  to  find  the 
other  parts. 

.        r  The  angle  ^  ==  30°  00',  the  angle  0  =  60°  00', 
^^'  \      and  AB  =  4178. 

13.  The  distance  between  the  earth  and  sun  is  91,500,000 
miles,  and  at  that  distani^.e  the  semi-diameter  of  the  snn 
subtends  an  angle  of  16'.  What  is  the  diameter  of  the 
sun  in  miles?  '  Ans.  887,674. 


In  this  example,  let  B  be  the  center  of  the  earth,  aS^  that  of  the 
Bun,  and  BB  a  tangent  to  the  sun's  surface.  Then  the  A  BBJS 
is  right-angled  at  B,  and  BS  is  the  semi-diameter  of  the  sun.  The 
«^lue  of  2BS  is  required. 


SECTION  II.  291 

14.  The  equatorial  diameter  of  the  earth  is  7925  miles, 
and  the  distance  of  the  sun  91,500,000  miles.  What  angle 
will  the  semi-diameter  of  the  earth  subtend,  as  seen  from 
the  sun  ?  Ans.  8.94". 

This  angle  is  called,  in  astronomy,  the  sun's  horizontal  parallax. 
The  preceding  figure  applies  to  this  example,  by  supposing  E  to 
be  the  center  of  the  sun,  S  that  of  the  earth,  and  BS  equal  to 
3956  miles. 

15.  The  mean  distance  of  the  moon  frv>m  the  earth  is 
60.3  times  3960  miles,  and  at  this  distance  the  semi- 
diameter  of  the  moon  subtends  an  angle  of  15'  32*'. 
VVhat  is  the  diameter  of  the  moon  in  miles  ? 

/-v  ^ns.  2157.8  miles. 

n.   OBLIQUE-ANGLED  TRIGONOMETRY. 

PROBLEM   I. 

Irt  a  plane  triangle,  given  a  side  and  the  two  adjacent 
angles,  to  find  the  other  parts. 

In  the  triangle  ABC,  let  AB  =  ^ 

376,  the  angle  A  =  48°  3',  and  the 
angle  B  =  40°  14',  to  find  the  other 
parts. 

As  the  sum  of  the  three  angles  of  every   ^  ^ 

triangle  is  always  180^,  the  third  angle,  C,  must  be  180°  —  88® 
17' =  91°  43'. 

To  find  AC. 

Sin.  91°  43',  9.999805 

:    ^5.376,  2.575188 

::  sin.B  4r°  U,  9.810167 


12.385355 


:   JC,  243,  2.385550 

Observe,  that  thb  sine  of  91°  43'  is  the  same  as  the  cosine  of 
1°43' 


21^2  PLANE    TRIGONOMETRY. 

To  find  BC. 

Sin.  91°  43',  9.999805 

:   ul5,  376,  2.575188 

: :  sin.  A,  48°  3',  9.871414 

12.446602 


:   sin.  5(7,  279.8,  2.446797 

PROBLEM   11. 

In  a  plane  triangle,  given  two  sides  and  an  angle  opposite 
one  of  them,  to  determine  the  other  parts. 

Let  AD  =  1751  feet,  one 
of  the  given  sides ;  the  angle 
D  =  31°  17'  19" ;  and  the  side 
opposite,  1257.5.  From  these 
data,  we  are  required  to  find 
the  other  side  and  the  other 
two  angles. 

In  this  case  we  do  not  know  whether  AG  or  AE  representa 
1257.5,  because  J.(7  =  AE.  If  we  take  AG  for  the  other  given 
side,  then  DG\^  the  other  required  side,  and  DA  G  is  the  vertical 
angle.  If  we  take  AE  for  the  other  given  side,  then  DE  is  the 
required  side,  and  DAE  is  the  vertical  angle.  In  such  cases  we 
determine  both  triangles. 

To  find  the  angle  E  ^   C. 
(Prop.  4.)  AG=  AE  ^Vlbl.b,    log.     3.099508 

:   i),  31°  17'  19",  sin.     9.715460 

::  AD,  1751,  log.     3.243286 

12.958746 


jy==  C,46°  18',  sin.    9.859238 

From  180°  take  46°  18',  and  the  remainder  is  the  angle  DUA 
:  133°  42'. 

The  angle     DAG  =^  AGE  —  D,  (Th.  11,  B.  I); 
that  is,         DAG  =  46°  18'  —  31°  17'  19"  =  15°  C  iV\ 
The  angles  D  and  E,  taken  from  180°,  give 
DAE  ==  102°  24'  41". 


SECTION    II.  2i>3 

To  find  DQ. 

Sin.  Z),  31°  17'  19",  log.  9.715460 

:    At\  1257.5,  log.  3.099508 

::  ^m.DAC  15<»  0'  41",  log.  0.413317 


12.512825 

•    Z>a,  626.86, 

2.797165 

To  find  BE. 

Sin.A31°iri7", 

9.715460 

,  :    AE,  1257.5, 

3.099508 

::  sXn.DAEy  102°  24' 41", 

9.989730 

13.089238 

!  VE,  2364.7,  3.373778 

IIevark. — To  make  the  triangle  possible,  ^(7 must  not  be  less  than 
AB  the  sine  of  the  angle  Z>,  when  DA  i»  made  radius. 

PROBLEM    III. 

In  any  plane  triangle^  given  two  sides  and  the  included 
angle,  to  find  the  other  parts. 

Let  AB  =  1751,  (see  last  figure),  BU  =  2364.5,  and 
the  included  angle  B  —  31°  17'  19".  "We  are  required 
to  find  AEy  the  angle  BAE,  and  the  angle  E. 

Observe  that  the  angle  E  must  be  less  than  the  angle  BAE,  bo- 
cause  it  is  opposite  a  less  ftide. 

From  180° 

Take  /),  31°  17'  19", 

Sum  of  the  other  two  angles,  =  148°  42'  41",  (Th.  11,  B.  I), 
J  sum  =     74°  21'  20". 

By  Proposition  7, 
DE+BA:  DE—  BA  -=  tan.  74°  21'  20"  :  tarn,  i(BAE^E\ 
That  is, 

4115.5  ;  613.5  «  tan.  74°  21' 20"  :  taiL.l{BAE—E) 
25* 


294 


PLANE    TRIGONOMETRY. 


Tan.  74°  21'  20'', 
613.5. 


4115.5  log.  (subtracted), 


10.552778 

2.787815 

13.340593 
3.614423 


tan.K^^^-^)  tan.28°  V  36",  9.726170 

15 ut  the  half  sum  plus  the  half  difference  of  any  two  quantities 
h  equal  to  the  greater  of  the  two;  and  the  half  sum  minus  the 
half  difference  is  equal  the  less. 


Therefore,  to 
Add 

DAE  =. 
E        = 


74°  21'  20", 
28°    r36", 

102°  22'  56", 
46°  19'  45", 


To  find  AE, 

Sin.  E,  46°  19'  45", 
:    DA,  1751, 
::  sin.A  31°  17' 19", 


AE,  1257.2, 


9.859323 
3.243286 
9.715460 

12.958746 

3.099423 


PROBLEM    IV. 
Given,  the  three  sides  of  a  plane  triangle,  to  find  the  angle$. 

Let  ^(7  =  1751,   CB  =  1257.5,  AB  =  2364.5,  to  find 
tlie  angles  A,  B,  and  0. 

II'  we  take  the  formula  for  cosines,  we  C 

will  compute  the  greatest  angle,  which  is 
0.     To  correspond  with  the  formula, 


cos.  i  C 


=  n/ 


Rh  (s  —  c) 


a6        '  A  B 

we  must  take     a  =--  1257.5,  h  =  1751,  and  c  =  2364.6. 
The  half  sum  of  these  is, 

s  =  2686.5;  and  s  —  c^  322. 


SECTION 

II. 

/?« 

20.000000 

8  =  2686.5 

3.429187 

«  —  c  =  322 

2.507856 

Numerator,  log. 

25.937043 

a  1257.5     3.099508 

h  1751.       3.243286 

Denominator,  log.    6.342794       6.342794 

2)19.594249 

K'=    51°  ir  10"    COS.  9.797124 
C=  102    22    20 

Tlie  remaining  angles  may  now  be  found  by  Problem  4. 


PRACTICAL    PROBLEMS. 

Let  ABC  represent  any  oblique-angled  triangle. 
1.  Given,  AB  697,  the  angle  A  81°  30'  10'',  and  the 
adgle  B  40°  30'  44",  to  find  the  other  parts. 

Ans,  AC,  534;  BC,  813;  and  L^,  57°  59'  6". 
•J.  K  AC  ^nO.S,  l_A  =  70°  5'  22",  L^  =  59°  35' 
36",  required  the  other  parts. 

Ans.  AB,  643.2;  BC,  785.8;  and  [_C,  50°  19'  2". 

3.  Given,  BC  980.1,  the  angle  A  7°  6'  26",  and  the 
angle  B  108°  2'  23",  to  find  the  other  parts. 

Ans.  AB,  7283.8;  AC,  7613.1 ;  and  L^,  66°  51'  11". 

4.  Given,  AB  896.2,  .5(7  328.4,  and  the  angle   (7113* 
15'  20",  to  find  the  other  parts. 

.(AC,  112;  L-i,19°35'46"> 
'  I     and  L^,  46°  38'  54".' 

5.  Given,  ^(7=  4627,  BC=  5169,  and  the  angle  A  =- 
Y0°  25'  12",  to  find  the  other  parts. 

.      1^5,4828;  L^,  57°  29' 56"; 
•  I      and  L^,  52°  4'  52". 


296  PLANE    TKIGUi\OMETE"k. 

6.  Given,  AB  793.8,  BO  481.6,  and  AC  500.0,  to  lina 
the  angles. 

,       ( l_A,  35°  15'  32'';   L-^,  36°  49'  18";   and   L^ 
*  I      107°  55'  10". 

7.  Given,  ^^  100.3,  BO  100.3,  and  ^(7  100.3,  lo  find 
the  angles. 

.       f  riie   angle  ^,  60°;   the  angle  B,  60°;   and  the 
^  ""  *  I      angle  6^,  60°. 

8.  Given,  AB  92.6,  BO  46.3,  and  ^6'  71.2,  to  lind  thf 
angles. 

,^     fL^,  29°17'22";    L^,  48°  47' 30";   and   L^, 
^""'•l      101°  55'  8". 

9.  Given,  AB  4963,  BO  5124,  and  AO  5621,  to  find 
the  angles. 

.       /L-4,  57°  30'28";    L_-^,  67°  42' 36";   and   L^, 
^'''- 1      54°  46'  55". 

10.  Given,  AB  728.1,  BO  614.7,  and  ^C  583.8,  to  find 
the  angles. 

.       f  L^  =  54°  32'  52",  L^=  50°  40'  58",  and  l_0 
^^'  \      =  74°  46'  10". 

11.  Given,  AB  96.74,  BO  83.29,  and  AO  111.42,  to 
find  the  angles. 

A      j  L^  =  46°  30'  45",    L^  =  76°  3'  46",  and   L^ 
^''''l      =57°  25'  29". 
^  12.  Given,  AB  363.4,  BO  USA,  and  the  angle  B  102° 
18'  27",  to  find  the  other  parts. 

.      j\__A  =  20°  9'  17",  the  side  A0  =  420.8,  and  [_0 
-^""''X      =57°  32'  16". 

1.3.  Given,  AB  632,  BO  494,  and  the  angle  A  20°  13', 
to  find  the  oth^r  parts,  the  angle  (7  being  acute. 

r  L^=  26°  18'  19",  L  ^  =  133°  25'  41",  and 
^^'•\      AO=10S5X 
14.  Given,  AB  53.9,  AO  46.21,  and  the  angle  B  58** 
16',  to  find  the  other  parts. 
Ans.  \_A  =  38°  58',  l_0=  82°  46',  and      B0-=  34.16. 


isECTlUiX    II.  297 

15.  Given,  AB  2163,  BC  1672,  and  the  angle  C  112° 
18'  22",  to  find  the  other  parts. 

Ans.  AC,  877.2;  L^^,  22°  2'  16";  and  L^,  45°  39'  22". 

16.  Given,  AB  496,  BO  496,  and  the  angle  B  38°  16', 
to  find  the  other  parts. 

Ans,  AO,  325.1;  L^,  70°  52';  and  L^,  70°  52'. 

17.  Given,  AB  428,  the  angle  0  49°  16',  and  {AC  + 
BC)  918,  to  find  the  other  parts,  the  angle  B  being 
obtuse. 

.       r  The  angle  A  =  38°  44'  48",  the  angle  B  =  91° 
'  ^*l      59' 12",  ^(7=564.5,   and  ^(7=353.5. 

18.  Given,  AC  126,  the  angle  B  29°  46',  and  {AB-^ 
^(7)  43,  to  find  the  other  parts. 

.      JThe  angle  ^  =  55°  51'  32",  the  angle  (7=94° 
^'  \      22'  28",  AB  =  253.05,  and  ^C  =  210.05. 

19.  Given,  AB  1269,  AC  1837,  and  the  angle  A  53*» 
16'  20",  to  find  the  other  parts. 

A       f  LJ^  =  83°  23'  47",  I    (7=  43°  19'  53",  and  BC 
\      « 1482.16. 


PLANE    TKIUONOMETRY. 


SECTION   III. 


A.PI  LIGATION  OF  TRIGONOMETRY  TO  MEASURING 
HEIGHTS  AND  DISTANCES. 

In  tliis  useful  application  of  Trigonometry,  a  base  line 
Is  always  supposed  to  be  measured,  or  given  in  length; 
and  by  means  of  a  qn  ad  rant,  sextant,  circle,  theodolite, 
or  some  other  instrument  for  measuring  angles,  such 
angles  are  measured  as,  connected  with  the  base  line  and 
the  objects  whose  heights  or  distances  it  is  proposed  to 
determine,  enable  us  to  compute,  from  the  principles  of 
Trigonometry,  what  those  heights  or  distances  are. 

Sometimes,  particularly  in  marine  surveying,  horizontal 
angles  are  determined  by  the  compass ;  but  the  varying 
effect  of  surrounding  bodies  on  the  needle,  even  in  situa- 
tions little  removed  Irom  each  other,  and  the  general 
construction  of  the  instrument  itself,  render  it  unfit  to  be 
employed  in  the  determination  of  angles  where  anything 
like  precision  is  required. 

The  following  problems  present  sufficient  variety,  to 
guide  the  student  in  determining  what  will  be  the  most 
eligible  mode  of  proceeding,  in  any  case  that  is  likely  to 
occur  in  practice. 

PROBLEM   I. 

Being  desirous  of  finding  the  distance  between  two 
listant  objects,  Q  and  Z),  I  measured  a  base,  AB,  of  384 
7ards,  on  the  same  horizontal  plane  with  the  objects  0 


SECTION    III.  299 

and  D,  At  A,  I  found  the  angles  DAB  =  48<*  12',  and 
CM^  =  89°  18';  at  B,  the  angles  ABC  46°  14',  and 
ABB  87°  4'.  It  is  required,  from  these  data,  to  com- 
pute the  distance  between  0  and  D, 

From  the  angle  CAB,  take  the  angle  DAB ;  the 
remainder,  41°  6*,  is  the  angle  CAD.  To  the  angle 
DBAj  add  the  angle  DAB,  and  44°  44',  the  supple- 
ment of  the  sum,  is  the  angle  ADB.  In  the  same 
way  the  angle  ACB,  which  is  the  supplement  of 
the  sum  of  CAB  and  CBA,  is  found  to  be  44°  28'. 

Hence,  in  the  triangles  ABC  and  ABD,  we  have 


Sin.  ACB,  44°  28', 
:   AB,  384  yards, 
::  Bin.  ABCy  46°  14', 

9.845405 
2.584331 

9.858635 

12.442966 

:   ACy  ^95.9  yards, 

2.597561 

Sin.  ADB,  44°  44', 
:   AB,  384  yards, 
::  sm.ABD,  87°  4', 

9.847454 
2.584331 
9.999431 

12.583762 

:   AD,  544.9  yards. 

2.736308 

Then,  in  the  triangle  CAD,  we  have  given  the  sides  GA  and  AD^ 
and  the  included  angle  CAD,  to  find  CD)  to  compute  which  we 
proceed  thus : 

The  supplement  of  the  angle  CAD,  is  the  sum  of  the  angles 
ACD  and^Z)C; 

Hence, =  69°  27';  and,  by  proportion  w«  have, 

AD-^-AG  (=-      940.8)        2.973497 

:  AD^AC  (=         149)        2.173186 

:;  faip/^^  +  ^^^     (=  69°  27')       10.426108 


12.599294 


300 


tan. 


PLANE   TRIGONOMETRY. 
ACD—ADG 


2 


(=  22°  54')        9.625797 


the  angle  A  CD,     sum,  92"  21' 
the  angle  ^2)  C,     diff.,  46°  33' 

Sm.J.i)(7,  46°83',    • 

:    ^(7,  395.9  yards, 
: :  sin.  CAD,  41°  6', 


CD,  358.5  yards, 


9.860922 
2.597585 
9.817813 

12.415398 

2.55447B 


PROBLEM    II. 

To  determine  the  altitude  of  a  lighthouse,  I  obsei-vtJ 
the  elevation  of  its  top  above  the  level  sand  on  the  sea- 
shore, to  be  15°  32'  18'' ;  and  measuring  directly  from 
it,  638  yards  along  the  sand,  I  then  found  its  elevation 
to  be  9°  56'  26".    Required  the  height  of  the  lighthouse 

Let  CD  represent  the  height  of  the  light- 
house above  the  level  of  the  sand,  and  let  B 
be  the  first  station,  and  A  the  second ;  then 
the  angle  CBD  is  15°  32'  18,  and  the  angle 
CAB  is  9°  56'  26";  therefore,  the  angle 
A  CB,  which  is  the  diflference  of  the  angles 
CBD  and  CAB,  is  5°  35'  52". 

Hence,  Sin.  ^  CjB,  5°  35' 52", 

:    AB,  638, 
: :  sin.  angle  A,  9°  56'  26", 


8.989201 
2.804821 
9.237107 


BCj  1129.09  yards, 

Radius, 

BC,  1129.09, 

sin.  CBD,  15°  32'  18", 


/)C;  302.46  yards. 


12  041928 
3.052727 

10.000000 
3.052727 
9.427945 

12.480672 

2.480672 


SECTION    III 


301 


PROBLEM    III. 

Coming  from  sea,  at  the  poiut  D  I  observed  two 
headlands,  A  and  B,  and  inland,  at  (7,  a  steeple,  whicb 
appeared  between  the  headlands.  I  found,  from  a  map, 
that  the  headlands  were  5.35  miles  apart ;  that  the  dis- 
tance from  A  to  the  steeple  was  2.8  miles,  and  from 
B  to  the  steeple  3.47  miles ;  and  I  found,  with  a  sextant, 
that  the  angle  J.i>(7  was  12°  15,  and  the  angle  BDQ,  15° 
30'.  Required  my  distance  from  each  of  the  headlands, 
and  from  the  steeple. 


CONSTRUCTION. 

The  angle  between  the  two  headlands  is 
the  sum  of  15°  30'  and  12°  15',  or  27°  45'. 
Take  double  this  sum,  55°  30'.  Conceive  AB 
to  be  the  chord  of  a  circle,  and  the  arc  on 
one  side  of  it  to  be  55°  30' ;  and,  of  course, 
the  other  will  be  304°  30'.  The  point  D 
will  be  somewhere  in  the  circumference  of 
this  circle.     Consider  that  point  as  determined,  and  draw  CD. 

In  the  triangle  ABC,  we  have  all  the  sides,  and,  of  course,  we 
can  find  all  the  angles ;  and  if  the  angle  A  CB  is  less  than  180°— 
27°  45'  =  152°  15',  then  the  circle  cuts  the  line  CD  in  a  point 
E,  and  C  is  without  the  circle. 

Draw  AEy  BE,  AD,  and  BD.  AEBD  is  a  quadrilateral  in  a 
circle,  and  [_  AEB  +  \_  ADD  =  180°. 

The  [^ADE=  the  | -4^^,  because  both  are  measured  by  one 

half  the  arc  AE.     Also,  [_  EDB  =  [__  EAB,  for  a  similar  reason. 

Now,  in  the  triangle  AEB,  its  side  AB,  and  all  its  angles,  are 
known;  and  from  thence  AE  can  be  computed.  Then,  having  the 
two  sides,  AC  and  AE,  of  the  triangle  AEC,  and  the  included 
angle  CAE,  we  can  find  the  angle  AEC,  and,  of  course,  its  supple- 
ment, AED.  Then,  in  the  triangle  AED,  we  have  the  side  AEy 
and  the  two  angles  AED  and  ADE,  from  which  we  can  find  AD 

The  computation,  at  length,  is  as  follows : 
26 


302 


PLANE    TRIGONOMiCTRY. 


To  find  AE. 

Angle  EaB  =     15<^  30'  Sin.  AEB,  152°  15',  9.668027 

Angle  fy^^  =     12°  15'      :    AB,  5.35,  .728354 

27°  45'     : :  sin.  ABE  12°  15'      9.326700 

180° 


AP«5l«  AEB  =  152°  15' 


AEy  2.438, 


To  find  the  angle  BAQ. 


10.055054 


.387027 


BG,    3.47 
AB,    5.35 

AGy    2.80 

log.     .728354 
log.     .447158 

2  )  11.62 

1.175512 

5.81 
^C,     2.34 

log.     .764176 
log.     .369216 
20 

21.133392 

17°  41'  58" 

2  )  19.957880 
COS.  9.978940 

2 

Angle  ^^C=.-  35°  23' 56" 
Angle  EAB  =  15°  30' 

Angle  EAG^  19°  53' 56" 

180° 

2)160°    6'    4" 
80°    3'    2" 

AEG  4-  AGE 

SECTION   III 


803 


To  find  the  angles  AEQ  and  ACE, 


%ng\QAEC, 


AC  -\-  AE 

:   AC^AE 

^      AEC -ir  AGE 
::tan.            ^ 

6.238 
.362 

80<»    3'    2" 
21°  30'  12" 

.719165 
—  1.558709 

10.755928 

10.314637 

,      AEC—ACE 
:   ten.            2 

9.595472 

7(7,              10P33a4^^8uin 

vngXeACEorACD,  58°32^50^^  diff. 
MiiileCDA,  12°  15' 


70°47'50'^  supplement  109°12a0'^  angle  CAD 
-   •   35°  23^  56'^  angle  CAB 


. 

73°48a4'^ 

To  find  AB. 

Sin.^Z>^,  12°15', 
:   ^(7,2.8, 
::  Bm.ACD  68°  32' 50", 

9.326700 

.447158 

9.930985 

10.378143 

•   AD  11.26  miles. 

1.051443 

PROBLEM    IV. 

The  elevation  of  a  spire  at  one  station  was  23°  50'  17  ', 
vid  the  horizontal  angle  at  this  station,  between  the 
spire  and  another  station,  was  93°  4'  20".  The  horizon- 
tal angle  at  the  latter  station,  between  the  spire  and  the 
first  station,  was  54°  28'  36",  and  the  distance  between 
the  two  stations  was  416  feet.  Required  the  height  ot 
the  spire. 


304 


PLANE    TRIGONOMETRY. 


Let  CD  be  the  spire,  A  the  first  station,  and 
B  the  second  ]  then  the  vertical  angle  CAD  is  y"^ 

23°  50'  17";  and  as  the  horizontal  angles,  CAB  ^  .-^^^  / 
and  CBA,  are  93°  4'  20"  and  54°  28'  36",  re-     \     / 
spectively,  the  angle  ACB,  the  supplement  of 
their  sura,  is  32°  27'  4". 


To  find  AC. 
Sin.  5(7^,  32°  27' 3", 
:    side  AB,  416, 
::  sin.^jBa,  54°28'36", 


side^C,  681, 


9.729G34 
2.619093 
9.910560 

12.529653 

2.800019 


To  find  DO. 

Radius,  10.000000 

side  AC,  631,  2.800019 

tan.  DA  (7,  23°  50'  17",  9.645270 


DC,  278.8, 


2.445289 


By  the  application  of  Pro- 
blem 4,  we  can  compute  the 
distance  between  two  horizon- 
tal planes,  if  the  same  object 
is  visible  from  both.  a^ 

For  example,  let  ilf  be  a 
prominent  tree  or  rock  near 

the  top  of  a  mountain,  and  by  observations  taken  at  A^ 
we  can  determine  the  perpendicular  Mn,  By  like  obser* 
vations  taken  at  B,  we  can  determine  the  perpendicular 
Mm,  The  difierence  between  these  two  perpendiculars  is 
nm,  or  the  difference  in  the  elevation  between  the  two 
points  A  and  B,  If  the  distances  between  A  and  n,  or  B 
and  m,  are  considerable,  or  more  than  two  or  three  miles, 
corrections  must  be  made  for  the  convexity  of  the  earth ; 
but  for  less  distances  such  corrections  are  not  necessaiy. 


SECTION    III.  306 

PRACTICAL    PROBLEMS. 

1.  Required  the  height  of  a  wall  whose  augle  of  eleva- 
tion, at  the  distance  of  463  feet,  is  observed  to  be  16° 
21'.  Ans.  135.8  feet. 

2.  The  angle  of  elevation  of  a  hill  is,  near  its  bottom, 
81°  18',  and  214  yards  farther  off,  26°  18'.  Required  the 
peq-jendieular  lieight  of  the  hill,  and  the  distance  of  the 
perpendicular  from  the  first  station. 

(  The  height  of  the  hill  is  565.2  yards,  and  the 
^ns.  <      distance  of  the  perpendicular  from  the  first 
I     station  is  929.6  yards. 

3.  The  wall  of  a  tower  which  is  149.5  feet  in  height, 
makes,  with  a  line  drawn  from  the  top  of  it  to  a  distant 
object  on  the  horizontal  plane,  an  angle  of  57°  21'. 
What  is  the  distance  of  the  object  from  the  bottom  of 
the  tower?  Ans.  233.3  feet. 

4.  From  the  top  of  a  tower,  which  is  138  feet  in  height, 
I  took  the  angle  of  depression  of  two  objects  standing 
in  a  direct  line  from  the  bottom  of  the  tower,  and  upon 
the  same  horizontal  plane  with  it.  The  depression  of  the 
nearer  object  was  found  to  be  48°  10',  and  that  of  the 
further,  18°  52'.  What  was  the  distance  of  each  from 
the  bottom  of  the  tower  ? 

4         f  Distance  of  the  nearer,  123.5  feet;  and  of  the 
\      further,  403.8  feet. 

5.  Being  on  the  side  of  a  river,  and  wishing  to  know 
the  distance  of  a  house  on  the  opposite  side,  I  measured 
312  yards  in  a  right  line  by  the  side  of  the  river,  and  then 
/bund  that  the  two  angles,  one  at  each  end  of  this  line, 
subtended  by  the  other  end  and  the  house,  were  31°  15' 
and  86°  27'.  What  vas  the  distance  between  each  end 
of  the  line  and  the  house  ?      Ans,  351.7,  and  182.8  yards. 

6.  Having  measured  a  base  of  260  yards  in  a  straight 
line,  on  one  bank  of  a  river,  I  found  that  the  two 
angles,  one  at  each  end  of  the  line,  subtended  by  the 

26*  U 


goo  PLANE    TKIGONOMETRY. 

other  end  and  a  tree  on  the  opposite  bank,  were  40®  and 
80°.     What  was  the  width  of  the  river? 

Ans.  190.1  yards. 

7.  From  an  eminence  of  268  feet  in  perpendicular 
height,  the  angle  of  depression  of  the  top  of  a  steeple 
which  stood  on  the  same  horizontal  plane,  was  found  to 
1  e  40^  8',  and  of  the  bottom,  06°  18'.  What  was  the 
height  of  the  steeple  ?  Ans.  117.76  feet. 

8.  Wanting  to  know  the  distance  between  two  objects 
which  were  separated  by  a  morass,  I  measured  the  dis- 
tance from  each  to  a  point  from  whence  both  could  be 
seen ;  the  distances  were  1840  and  1428  yards,  and  the 
angle  which,  at  that  point,  the  objects  subtended,  was  36** 
18' 24''.    Required  their  distance.     J.n«.  1090.85  yards. 

9.  From  the  top  of  a  mountain,  three  miles  in  height, 
the  visible  horizon  appeared  depressed  2°  13'  27".  Re- 
quired the  diameter  of  the  earth,  and  the  distance  of  the 
boundary  of  the  visible  horizon. 

.      .         r  Diameter  of  the  earth,  7958  miles ;  distance  of 
\      the  horizon,  154.54  miles. 

10.  From  a  ship  a  headland  was  seen,  bearing  north 
39°  23'  east.  After  sailing  20  miles  north,  47°  49'  west, 
the  same  headland  was  observed  to  bear  north,  87°  11' 
east.  Required  the  distance  of  the  headland  from  the 
ship  at  each  station. 

J        J  At  first  station,  19.09  miles ;   at  the  second, 
^''*-   I      26.96  miles. 

11.  The  top  of  a  tower,  100  feet  above  the  level  of  tlie 
sea,  was  seen  as  on  the  surface  of  the  sea,  from  the  mast- 
head of  a  ship,  90  feet  above  the  water.  The  diameter 
of  the  earth  being  '''960  miles,  what  was  the  distance 
between  the  observei  and  the  object? 

Ans.  23.92  plus  j\  for  refraction  —  25.76  miles. 

12.  From  the  top  of  a  tower,  by  the  seaside,  143  feet 
high,  it  was  observed  that  the  angle  of  depression  of  a 


SECTION   III.  307 

ship's  bo'tom,  then  at  anchor,  measured  35®  ;  what,  then, 
was  the  ".hip's  distance  from  the  foot  of  the  tower  ? 

Ans.  204.22  feet. 

13.  Wanting  to  know  tlie  breadth  of  a  river,  I  meas- 
ured a  base  of  500  yards  in  a  straight  line  on  one  bank; 
and  at  each  end  of  this  line  I  found  the  angles  subtended 
l)y  the  other  end  and  a  tree  on  the  opposite  bank  of  the 
river,  to  be  53°  and  79°  12'.  What,  then,  was  the  per- 
pendicular breadth  of  the  river?         Ans.  529.48  yards. 

14.  What  is  the  perpendicular  height  of  a  hill,  ita 
augle  of  elevation,  taken  at  the  bottom  of  it,  being  46°, 
and  200  yards  further  off,  on  a  level  with  the  bottom, 
81°  ?  Ans.  286.28  yards. 

15.  Wanting  to  know  the  height  of  an  inaccessible 
tower,  at  the  least  accessible  distance  from  it,  on  the 
same  horizontal  plane,  I  found  its  angle  of  elevation  to 
be  58° ;  then  going  300  feet  directly  from  it,  I  found  the 
angle  there  to  be  only  32° ;  required  the  height  of  the 
tower,  and  my  distance  from  it  at  the  first  station. 

307.54  feet. 


f  Height, 
\  Distance, 


^ns ^32.18    " 


16.  Two  ships  of  war,  intending  to  cannonade  a  fort, 

are,  by  the  shallowness  of  the  water,  kept  so  far  from  it, 

that  they  suspect  their  guns  cannot  reach  it  with  effect. 

In  order,  therefore,  to  measure  the  distance,  they  separate 

fi'om  each  other  a  quarter  of  a  mile,  or  440  yards,  and  then 

each  ship  observes  and  measures  the  angle  wliich  the 

other  ship  and  fort  subtends ;  these  angles  are  83°  45', 

and  85°  15'.     What,  then,  is  the  distance  between  each 

ship  and  the  fort?  .        /  2292.26  yards. 

^ns.  I  2298.05      " 

17.  A  point  of  land  was  observed  by  a  ship,  at  sea,  to 
bear  east-by-south  ;*  and  after  sailing  north-east  12  miles, 

*  That  is,  one  point  south  of  east.     A  point  of  the  compass  ii 


308  PLANE    TRIGONOMETRY. 

it  was  found  to  bear  south- cast-by- east.  It  is  required  to 
determiue  the  place  of  that  headland,  and  the  ship's  dis- 
tance from  it  at  the  last  observation. 

Ans,  Distance,  26.0728  miles. 

18.  Wishing  to  know  my  distance  from  an  inaccessible 
object,  0,  on  the  opposite  side  of  a  river,  and  having 
a  chain  or  chord  for  measuring  distances,  but  no  instru- 
ment for  taking  angles;  from  each  of  two  stations,  A 
and  B,  which  were  taken  at  500  yards  asunder,  I  meas- 
ured in  a  direct  line  from  the  object,  0,  100  yards,  viz., 
A.0  and  BB,  each  equal  to  100  yards;  and  I  found  that 
the  diagonal  AB  measured  550  yards,  and  the  diagonal 
BO  560.  What,  then,  was  the  distance  of  the  object  0 
from  each  station  A  and  Bt     .        (  AO,  536.27  yards. 

^^^'  \J50, 500.14     " 

19.  A  navigator  found,  by  observation,  that  the  summit 
of  a  certain  mountain,  which  he  supposed  to  be  45  min- 
utes of  a  degree  distant,  had  an  altitude  above  the  sea 
horizon  of  31'  20".  Now,  on  the  supposition  that  the 
earth's  radius  is  3956  miles,  and  the  observer's  dip  was 
4'  15",  what  was  the  height  of  the  mountain  ? 

Ans.  3960  feet. 

Remark.  —  This  should  be  diminished  by  about  one  eleventh 
part  of  itself,  for  the  influence  of  horizontal  refraction. 

20.  From  two  ships,  A  and  B,  which  are  anchored  in 
a  bay,  two  objects,  (7  and  i),  on  the  shore,  can  be  seen. 
These  objects  are  known  to  be  500  yards  apart.  At  the 
ship  A,  the  angle  subtended  by  the  objects  was  measured, 
and  found  to  be  41°  25' ;  and  that  by  the  object  B  and 
the  other  ship  was  found  to  be  52°  12'.  At  the  other 
ship,  the  angle  subtended  by  the  objects  on  shore  was 
found  to  be  48°  10';  and  that  by  the  object  (7,  and  the 
ship  A,  to  be  47°  40'.     Eequired  the  distance  between 


SECTION    III.  309 

tJie  ships,  and  the  distance  from  each  ship  to  the  objecta 

on  shore. 

{Distance  between  sliips,     395.7  yards. 
From  ship  A  to  object  i),  743.5      " 
From  ship  A  to  object  0,  407.7     " 
From  ship  B  to  object  D,  590.5      " 

To  solve  this  problem,  suppose  the  distance  between  the  ships  ta 
be  100  yards,  and  determine  the  several  distances,  including  the 
distance  between  the  objects,  C  and  />,  under  this  supposition;  then 
multiply  the  values  thus  found  for  the  required  distances  by  the 
quotient  obtained  by  dividing  the  given  value  of  CD  by  the  com- 
puted value. 


PART    II. 

SPHEEICAL   GEOMETRY 

AND 

TRIGONOMETRY. 


SECTION  I. 

SPHERICAL   GEOMETRY. 

DEFINITIONS. 

1.  Spherical  Geometry  has  for  its  object  the  investiga- 
tion of  the  properties,  and  of  the  relations  to  each  other, 
of  the  portions  of  the  surface  of  a  sphere  which  are 
bounded  by  the  arcs  of  its  great  circles. 

2.  A  Spherical  Polygon  is  a  portion  of  the  surface  of  a 
sphere  bounded  by  three  or  more  arcs  of  great  circles,  called 
the  sides  of  the  polygon. 

3.  The  Angles  of  a  spherical  polygon  are  the  angles 
formed  by  the  bounding  arcs,  and  are  the  same  as  the 
angles  formed  by  the  planes  of  these  arcs. 

4.  A  Spherical  Triangle  is  a  spherical  polygon  having 
hut  three  sides,  each  of  which  is  less  than  a  semi-circum- 
ference. 

5.  A  Lune  is  a  portion  of  the  surface  of  a  sphere  in- 
chided  between  two  great  semi-circumferences  having  a 
common  diameter. 

6.  A  Spherical  Wedge,  or  Ungula,  is  a  portion  of  the 
solid  sphere  included  between  two  great  semi-circles  having 
a  common  diameter. 


feECTION    I.  311 

7.  A  Spherical  Pyramid  is  a  portion  of  a  sphere  bounded 
by  the  faces  of  a  solid  angle  having  its  vertex  at  the 
center,  and  the  spherical  polygon  which  these  faces  inter- 
cept on  the  surface.  This  spherical  polygon  is  called  the 
base  of  the  pyramid. 

8.  The  Axis  of  a  great  circle  of  a  sphere  is  that  diameter 
of  the  sphere  which  is  perpendicular  to  the  plane  of  the 
circle.  This  diameter  is  also  the  axis  of  all  small  circles 
parallel  to  the  great  circle. 

9.  A  Pole  of  a  circle  of  a  sphere  is  a  point  on  the  sur- 
face of  the  sphere  equally  distant  from  every  point  in  the 
circumference  of  the  circle. 

10.  Supplemental,  or  Polar  Triangles,  are  two  triangles  on 
a  sphere,  so  related  that  the  vertices  of  the  angles  of 
cither  triangle  are  the  poles  of  the  sides  of  the  other 

PROPOSITION   I. 

Ani/  two  sides  of  a  spherical  triangle  are  together  greater 
than  the  third  side. 

Let  AB,  AC,  and  BO,  be  the  three 
Bides  of  the  triangle,  and  2)  the  center 
of  the  sphere. 

The  angles  of  the  planes  that  form 
the  solid  angle  at  D,  are  measured  by 
the  arcs  AB,  AC,  awdBC.  But  any 
two  of  these  angles  are  together  greater 
than  the  third  angle,  (Th.  18,  B.  VI).  There  lore,  any  two 
Hides  of  the  triangle  are,  together,  greater  thau  the  third  side. 

Hence  the  proposition. 

PROPOSITION    II. 

TJie  sum  of  the  three  sides  of  any  spherical  triingle  is  le9% 
than  the  circumference  of  a  great  circle. 

Let  ABC  be  a  spherical  triangle ;  the  two  si.les,  AB 
and  AC,  produced,  will  meet  at  the  point  which  is  liarae- 
tncaUy  opposite  to  A,  and  the  arcs,  ABB  an-'  ACD  are 


312 


SPHERICAL    GEOMETRY. 


together  equal  to  a  great  circle.  But, 
by  the  last  proposition,  5(7  is  less 
than  the  two  arcs,  BD  and  D C.  There- 
fore, AB  -\-  BQ  +  AO,  is  less  than 
ABD  +  AOB'y  that  is,  less  than  a 
gi-eat  circle. 

Hence  the  proposition. 

PROPOSITION    III. 

The  extremities  of  the  axis  of  a  great  circle  of  a  sphere 
are  the  poles  of  the  great  circle,  and  these  points  are  also 
the  poles  of  all  small  circles  parallel  to  the  great  circle. 

Let  0  be  the  center  of 
the  sphere,  and  BB  the 
axis  of  the  great  circle. 
Cm  Am''-,  then  will  5  and 
2>,  the  extremities  of  the 
axis,  be  the  poles  of  the 
circle,  and  also  the  poles 
of  any  parallel  small  cir- 
cle, as  FnE, 

For,  since  BB  is  per- 
pendicular to  the  plane 
of  the  circle. 


Cm  Am",  it 


is  perpendicular  to  the  lines  OA,  Om\  Om",  etc.,  passing 
through  its  foot  in  the  plane,  (Def.  2,  B.  VI);  hence,  all 
the  arcs,  Bm,  Bm\  etc.,  are  quadrants,  as  are  also  the 
arcs  Bm,  Bm\  etc.  The  points  B  and  B  are,  therefore, 
each  equally  distant  from  all  the  points  in  the  circumfer- 
ence, Cm  Am";  hence,  (Def.  9),  they  are  its  poles. 

Again,  since  the  radius,  OB,  is  perpendicular  to  the 
Diane  of  the  circle.  Cm  Am",  it  is  also  perpendicular  to 
the  plane  of  the  parallel  small  circle,  FnE,  and  passes 
through  its  center,  0\  !N'ow,  the  chords  of  the  arcs,  BF, 
Bn,  BE,  etc.,  being  oblique  lines,  meeting  the  plane  of 
the  small  circle  at  equal  distances  from  the  foot  of  the 


SECTION    I.  313 

perpendicular,  BO'^  are  all  equal,  (Th.  4,  B.  VI);  Leuce, 
the  arcs  themselves  are  equal,  and  B  is  one  pole  of  the 
circle,  F71E.  In  like  manner  we  prove  the  arcs,  DF^  Dn, 
DF,  etc.,  equal,  and  therefore  I)  is  the  other  pole  of  the 
same  circle. 

Hence  the  proposition,  etc. 

Oor.  1.  A  point  on  the  surface  of  a  sphere  at  the  distance 
of  a  quadrant  from  two  points  in  the  arc  of  a  great  circle,  not 
at  the  extremities  of  a  diameter,  is  a  pole  of  that  arc. 

P^or,  if  the  arcs,  Bm,  Bm\  are  each  quadrants,  the  angles, 
BOm  and  BOm',  are  each  right  angles;  and  hence,  BO 
is  perpendicular  to  the  plane  of  the  lines,  Om  and  Om', 
which  is  the  plane  of  the  arc,  mm'-,  B  is  therefoi'e  the 
pole  of  this  arc. 

Cor.  2.  The  angle  included  between  the  arc  of  a  great  circle 
and  the  arc  of  another  great  circle,  connecting  any  of  its  points 
with  the  pole,  is  a  right  angle. 

For,  since  the  radius,  BO,  is  perpendicular  to  the  plane 
of  the  circle.  Cm  Am",  every  plane  passed  through  this 
radius  is  perpendicular  to  the  plane  of  the  circle ;  hence, 
the  plane  of  the  arc  Bm  is  perpendicular  to  that  of  the 
arc  Om;  and  the  angle  of  the  arcs  is  that  of  their  planes. 

PROPOSITION    IV. 

The  angle  formed  hy  two  arcs  of  great  circles  which  inter- 
sect each  other,  is  equal  to  the  angle  included  between  the  tan- 
gents to  these  arcs  at  their  point  of  intersection,  and  is  meas- 
ured by  that  arc  of  a  great  circle  whose  pole  is  the  vertex  of 
the  angle,  and  which  is  limited  by  the  sides  of  the  angle  or 
the  sides  produced. 

Let  AM  and  AN  be  two  arcs  intersecting  at  the 
p(»int  A,  and  let  AE  and  AF  be  the  tangentj-  to  these 
arcs  at  this  point.  Take  AC  and  AD,  each  quadrants, 
and  draw  the  arc  CD,  of  which  A  is  the  pole,  and  OC 
and  OD  are  the  radii. 
27 


814 


SPHERICAL   GEOMETRY. 


Now,  since  the  planes  of  the  arcs  intersect  in  the  radiui 
OA,  and  AU  is  a  tangent  to  one  arc,  and  AF  a  tangent 
to  the  other,  at  the  common  point  A,       ^  p 

these  tangents  form  with  each  other  an 
angle  whicii  is  the  measure  of  the  angle 
of  the  planes  of  the  arcs ;  bnt  the  angle 
of  the  planes  of  the  arcs  is  taken  as  the 
angle  included  by  the  arcs,  (Def.  3). 

Again,  because  the  arcs,  A  0  and  AD, 
are  each  quadrants,  the  angles,  A  00, 
A  OB,  are  right  angles ;  hence  the  radii, 
(9^  and  OD,  which  lie,  one  in  one  face, 
and  the  other  in  the  other  face,  of  the 
diedral  angle  formed  by  the  planes  of  the  arcs,  are 
perpendicular  to  the  common  intersection  of  these  faces 
at  the  same  point.  The  angle,  00 D,  is  therefore  the 
angle  of  the  planes,  and  consequently  the  angle  of  the 
arcs  ;  but  the  angle  OOD  is  measured  by  the  arc  OB. 

Hence  the  proposition. 

Oor.  1.  Since  the  angles  included  between  the  arcs  oi 
great  circles  on  a  sphere,  are  measured  by  other  arcs  of 
great  circles  of  the  same  sphere,  we  may  compare  such 
angles  with  each  other,  and  construct  angles  equal  to 
other  angles,  by  processes  which  do  not  differ  in  principle 
from  those  by  which  plane  angles  are  compared  and  con- 
structed. 

Oor.  2.  Two  arcs  of  great  circles  will  form,  by  their  in- 
tersection, four  angles,  the  opposite  or  vertical  ones  of 
"  which  will  be  equal,  as  in  the  case  of  the  angles  formed 
by  tte  intersection  of  straight  lines,  (Th.  4,  B.  I). 


PROPOSITION  V. 

The  surface  of  a  hemisphere  may  he  divided  into  three  right- 
angled  and  four  quadrantal  triangles,  and  one  of  these  right- 
angled  triayigles  will  be  so  related  to  the  other  two,  that  two 
of  its  sides  and  one  of  its  angles  will  he  complemental  to  th^ 


SECTION    I.  315 

itdes  of  one  of  them,  and  two  of  its  sides  supplemental  to  two 
of  the  sides  of  the  other. 

Let  ABC  be  a  right-aDgled  spherical  triangle,  right 
angled  at  B. 

Produce  the  sides,  AB  and  AC,  and 
they  will  meet  at  A',  the  opposite 
]»oint  on  the  sphere.  Produce  BC, 
both  ways,  90°  from  the  point  B,  to 
P  and  P',  which  are,  therefore,  poles 
to  the  arc  AB,  (Prop.  3).  Through 
A,  P,  and  the  center  of  the  sphere, 
pass  a  plane,  cutting  the  sphere  into 
two  equal  parts,  forming  a  great  circle  on  the  sphere, 
which  great  circle  will  be  represented  by  the  circle 
PAF'A^  in  the  figure.  At  right  angles  to  this  plane, 
pass  another  plane,  cutting  the  sphere  into  two  equal 
parts ;  this  great  circle  is  represented  in  the  figure  by  the 
straight  line,  POP',  A  and  A'  are  the  poles  to  the  great 
circle,  POP'  \  and  P  and  P'  are  the  poles  to  the  great 
circle,  ABA'. 

Now,  CPI)  is  a  spherical  triangle,  right-angled  at  i>, 
and  its  sides  OP  and  CD  are  complemental  respectively 
to  the  sides  BC  and  AC  of  the  A  ABC,  and  its  side  PB 
is  complemental  to  the  arc  BO,  which  measures  the 
[_BACof  thejsame  triangle.  Again,  the  A  A' BC  is  right- 
angled  at  P,  and  its  sides  A'C,  A'B,  are  supplemental 
respectively  to  the  Bides  AC,  AB,  of  the  a  ABC.  There- 
fore, the  three  right-angled  A's,  ABC,  CPD,  and  A'BC, 
Lave  the  required  relations.  In  the  A  ACP,  the  side  AP 
is  a  quadrant,  and  for  this  reason  the  A  is  called  a  quad- 
rantal  triangle.  So  also,  are  the  A's  A'CP,  ACP',  and 
P'CA',  quadrantal  trianficies.     Hence  the  proposition. 

Scholium. — In  every  triangle  there  are  six  elements,  three  sides  and 
tliree  angles,  caLed  the  parts  of  the  triangle. 

Now,  if  all  the  parts  of  the  triangle  ^^Care  known,  the  parts  of 
each  of  th«  ^*s,  PCD  and  A^BC,  are  as  completely  known.  And 
vhen  the  parts  ol  the  ^  PCD  are  known,  the  parts  of  the  A'^  ACP 


316 


SPHBRICAL    TRIGONOMETRY. 


and  A^CP  are  also  known  ;  for,  the  side  PD  measures  each  .>f  the  | ^^'i 

P4(7and  PA'C,  and  the  angle  CPD,  added  to  the  right  angle  A' PD, 

gives  the  | A^PC^  and  the  | CPA  is  supplemental  to  this.     Hence, 

the  solution  of  the  A  ABC'is  a  solution  of  the  two  right-angled  and 
four  qaadrantal  A's,  which  together  with  it  make  up  the  surface  of 
the  hemisphere. 

PROPOSITION  VI. 

If  there  he  three  ares  of  great  circles  whose  poles  are  the 
angular  points  of  a  spherical  triangle,  such  arcs^  if  produced^ 
VJill  form  another  triangle,  whose  sides  will  he  supplemental 
to  the  angles  of  the  first  triangle,  and  the  sides  of  the  first 
triangle  will  he  supplemental  to  the  angles  of  the  second. 

Let  the  arcs  of  the  three  great  cir- 
cles be  CrH,  PQ,  KL,  whose  poles  are 
respectively  A,  B,  and  Q.  Produce  the 
three  arcs  until  they  meet  in  I),  E,  and 
F.  We  are  now  to  prove  that  E  is  the 
pole  of  the  arc  AQ\  I)  the  pole  of  the 
arc  BQ\  F  the  pole  to  the  arc  AB. 
Also,  that  the  side  EF,  is  supplemental 
to  the  angle  A ;  EB  to  the  angle  (7; 
and  BF  to  the  angle  B;  and  also,  that  the  side  AC  i» 
supplemental  to  the  angle  E,  etc. 

A  pole  is  90°  from  any  point  in  the  circumference  of 
Hs  great  circle ;  and,  therefore,  as  A  is  the  pole  of  the 
arc  Gff,  the  point  A  is  90°  from  the  point  E.  As  O  is 
the  pole  of  the  arc  LK,  0  is  90°  from  any  point  in 
that  arc;  therefore,  (7  is  90°  from  the  point  E',  and 
E  being  90°  from  both  A  and  (7,  it  is  the  pole  of  the  arc 
AC  In  the  same  manner,  we  may  prove  that  B  is  the 
pole  of  BC,  and  F  ihQ  pole  of  AB. 

Because  A  is  the  pole  of  the  arc  GH,  the  arc  G H 
measures  the  angle  A,  (Prop.  4) ;  for  a  similar  reason, 
PQ  measures  the  angle  B,  and  LK  measures  the  angle  (7. 

Because  E  is  the  pole  of  the  arc    AQ.     EH=z  90° 

Or,  EG+  GH=^  90° 

For  a  like  reason,  FH^  GH-^  90° 


SECTION  I.  ai7 

Ailding  tliese  two  equations,  and  observing  that  Gfl 
mm  A,  and  afterward  transposing  one  A^  we  have, 
EG  4-  GH  +  FH^^nO''  —A. 

Or,  ^JP=180°— ^  ^ 

In  like  manner,  i^i)  =  180°— ^    \     («) 

And,  BE  =  180°  —  (7  J 

But  the  arc  (180°  — A),  is  a  supplemental  arc  to  -A,  by 
the  definition  of  arcs;  therefore,  the  three  sides  of  the 
triangle  DEF^  are  supplements  of  the  angles  A,  B,  C,  of 
the  triangle  ABC. 

Again,  as  E  is  the  pole  of  the  arc  AC,  the  whole  angle 
E  is  measured  by  the  whole  arc  LH. 

But,  AC -{- CIT ^  90"* 

Also,  AC +  AL  =  90^ 

By  addition,  AC+AC-\-CH  -\-  AL  =  180° 
By  transposition,  AC-\-Cff+AL  =  lSO°—AC 
That  is,  LR,  or  ^=  180°  — ^1(7  ^ 

In  the  same  manner,  ^=-180°— yl5    >  (^) 

And,  I)=1S0°  —  BC  J 

That  is,  the  sides  of  the  first  triangle  are  supplemental 
to  the  angles  of  the  second  triangle. 


PROPOSITION   VII. 

The  sum  of  the  three  angles  of  any  spherical  triangle,  i% 
greater  than  two  right  angles,  and  less  than  six  right  angles. 

Add  equations  ( « ),  of  the  last  proposition.  The  first 
member  of  the  equation  so  formed  will  be  the  sum  of 
the  three  bides  of  a  spherical  triangle,  which  sum  we 
may  designate  by  S.  The  second  member  will  be  6  right 
angles  (there  being  2  right  angles  in  each  180°)  less  the 
three  angles  A,  B,  and  C, 

That  is,  aS'  =  6  right  angles  —  {A -^  B  +  C) 

By  Prop.  2,  the  sum  S!  is  less  than  4  right  angles; 
27* 


318 


SPHERICAL    GEOMETRY. 


tlierelbre,  to  it  add  s,  a  sufficieut  quantity  to  make  4 
right  angles.     Then, 

4  right  angles  =  6  right  angles  —  (A  +  B  -^  0)  -\-  s 

Drop  or  cancel  4  right  angles  from  both  members,  and 

transpose  {A  -h  B  -h  Cf). 

Then,  A  +  B  +  0  =  2  right  angles  -f  8. 

That  is,  the  three  angles  of  a  spherical  triangle  mako 
a  greater  sum  than  two  right  angles  by  the  indefinite 
quantity  s,  which  quantity  is  called  the  spherical  excess, 
and  is  greater  or  less  according  to  the  size  of  the  triangle. 

Again,  the  sum  of  the  angles  is  less  than  6  right  angles 
There  are  but  three  angles  in  any  triangle,  and  each  one  of 
them  must  be  less  than  180°,  or  2  right  angles.  For,  an 
angle  is  the  inclination  of  two  lines  or  two  planes ;  and 
when  two  pianos  incline  by  180°,  the  planes  are  parallel, 
or  are  in  one  and  the  same  plane ;  therefore,  as  neither 
angle  can  be  equal  to  2  right  angles,  the  three  can  never 
be  equal  to  6  right  angles. 

PROPOSITION    VIII. 


On  the  same  sphere,  or  on  equal  spheres,  triangles  which 
are  mutually  equilateral  are  also  mutually  equiangular ;  and, 
conversely,  triangles  which  are  mutually  equiangular  are  also 
mutually  equilateral,  equal  sides  lying  opposite  equal  angles. 

First.— Lai  ABO  and  DEF,  in 
which  AB  =  BE,  AC=:^  DF,  and 
BO  =  EF,  be  two  triangles  on 
the  sphere  whose  center  is  0; 
then  will  the  [_  A,  opposite  the 
side  BO,  in  the  first  triangle,  be 
equal  the  \__I),  opposite  the  equal 
side  EF,  in  the  second;  also 
L  5  =  1    ^,  andL^==L-^. 


SECTION    I.  819 

For,  drawing  the  radii  to  tlie  vertices  of  the  angles  of 
these  triangles,  we  may  conceive  0  to  he  the  common 
vertex  of  two  triedral  angles,  one  of  which  is  hounded 
by  the  plane  angles  AOB,  BOO,  and  AOO,  and  the  other 
by  the  plane  angles  DOE,  EOF,  and  DOF.  But  the 
plane  angles  bounding  the  one  of  tliese  triedral  angles, 
arc  equal  to  the  plane  angles  hounding  the  other,  each 
t')  ea  h,  since  they  are  measured  by  the  equal  sides  of  the 
two  triangles.  The  planes  of  the  equal  arcs  in  the  two 
triangles  are  therefore  equally  inclined  to  each  other, 
(Th.  20,  B.  VI) ;  but  the  angles  included  between  the 
planes  of  the  arcs  are  equal  to  the  angles  formed  by  the 
i*rcs,  (Def  3). 

Hence  the  [_  A,  opposite  the  side  BC,  in  the  A  Al 
IS  equal  to  the  [_  ^?  opposite  the  equal  side  EF,  in  tl  ^ 
other  triangle  ;  and  for  a  similar  reason,  the  L_jB=  L_^, 

andtheL^(7=LJ^. 

Second. — If,  in  the  triangles  ABC  and  DEF,  being  f 
the  same  sphere  whose  center  is  0,  the  [_A  =  [_D,ih 
\_B  =  [_E,  and  the  [_C  =  [_F\  then  wall  the  side  A  B, 
opposite  the  \_C,\n  the  first,  be  equal  to  the  side  DE^ 
opposite  the  equal  \^F,  in  the  second;  and  also  the  si'* 
AC  equal  to  the  side  DF,  and  the  side  BQ  equal  to  tbc 
side  EF. 

For,  conceive  two  triangles,  denoted  by  A'B'C*  and 
D'E'F',  supplemental  to  ABC  and  DEF,  to  be  formed; 
then  will  these  supplemental  triangles  be  mutually  equi- 
lateral, for  their  sides  are  measured  by  180*^  less  the 
o]  posite  and  equal  angles  of  the  triangles  ABC  and 
LEF,  (Prop.  G);  and  being  mutually'  equilateral,  they 
are,  as  proved  above,  mutually  equiangular.  But  the 
triangle;!  ABC  and  DEF  are  supplemental  to  the  tri- 
argles  A*B'C'  and  D'E'F' ;  and  their  sides  are  therefore 
measured  severally  by  180°  less  the  opposite  and  equal 
aoifles  of  the  triangles  A'B'C  and  D'E'F\  (Prop.  6). 


320  SPHERICAL    GEOMETRY. 

Heuce  the  triangles  ABO  and  DUF,  which  are  mutUiiUy 
equiangular,  are  also  mutually  equilateral. 

Scholium. — With  the  three  arcs  of  great  circles,  AB,  AC,  and  BC 
either  of  the  two  triangles,  ABC,  DEF,  may  be  formed ;  but  't  is  evi 
dent  that  these  two  triangles  cannot  be  made  to  coincide,  tliough  they 
ars  both  mutually  equilateral  and  mutually  «quiangular.  Spherical 
triangles  on  the  same  sphere,  or  on  equal  spheres,  in  which  the  side«< 
and  angles  of  the  one  are  equal  to  the  sides  and  'ingles  of  the  other 
eaoii  to  each,  but  are  not  themselves  capable  of  superposition,  dirt 
called  symmetrical  triangles. 


PROPOSITION    IX. 

On  the  same  sphere,  or  on  equal  spheres,  triangles  having 
two  sides  of  the  one  equal  to  two  sides  of  the  other,  each  to 
each,  and  the  included  angles  equal,  have  their  remaining 
sides  and  angles  equal. 

Let  ABQ  and  DEF  be  two 
triangles,  in  which  AB  =^  BE, 
AC  =  DF,  and  the  angle  A  — 
the  angle  D;  then  will  the  side 
BC  he  equal  to  the  side  FF, 
the  L  ^  =  the  L^,  and  L  0 
^[_F 

For,  if  BF  lies  on  the  same 
side  of  BF  that  AB  does  of  AC,  the  two  triangles,  ABO 
and  BFF,  may  be  applied  the  one  to  the  other,  and  they 
may  be  proved  to  coincide,  as  in  the  case  of  plane  tri- 
angles. But,  if  BF  does  not  lie  on  the  same  side  of  BF 
that  AB  does  of  AC,  we  may  construct  the  triangle  "^  hicb 
is  symmetrical  with  BFF;  and  this  symmetrical  triangle, 
when  applied  to  the  triangle  ABC,  will  exactly  coincide 
with  it.  But  the  triangle  BFF,  and  the  triangle  sym- 
metrical with  it,  are  not  only  mutually  equilateral,  but 
also  are  mutually  equiangular,  the  equal  angles  lying 
opposite  the  equal  sides,  (Prop.  8) ;  and  as  the  one  or  the 
other  will  coincide  with  the  triangle  ABC,  it  follows  that 


^^£CTION    I.  321 

the  triangles,  ABC  and  DEF,  are  either  absolutely  or 
By m metrically  equal. 

Cor,  On  the  same  sphere,  or  on  equal  spheres,  trianglei 
having  two  angles  of  the  one  equal  to  two  angles  of  the  other^ 
each  to  each,  and  the  included  sides  equal,  have  their  remain- 
ing sides  and  angles  equal. 

For,  if  L^  =  L  A  L^  =  L^j  and  side  AB  =  side 
DE,  the  triangle  BUF,  or  the  triangle  symmetrical  with 
\t,  will  exactly  coincide  with  A  ABC,  when  applied  to  it 
as  in  the  case  of  plane  triangles;  hence,  the  sides  and 
angles  of  the  one  will  be  equal  to  the  sides  and  angles 
of  the  other,  each  to  each. 


PROPOSITION   X. 

In  an  isosceles  spherical  triangle,  the  angles  opposite  the 
equal  sides  are  equal, 

A 

Let  ABC  be  an  isosceles  spherical  tri- 
angle, in  which  AB  and  AC  are  the  equal 
sides ;  then  will  [_B  =  [_  (7. 

For,  connect  the  vertex  A  with  D,  the  / 

middle  point  of  the  base,  by  the  arc  of  a  / 
great  circle,  thus  forming  the  two  mutu-  JL^  I — ^ 
ally  equilateral  triangles,  ABB  and  ABC. 
They  are  mutually  equilateral,  because  AB  is  common, 
BB  =  DC hy  construction,  and  AB=AChy  supposition; 
hence  they  are  mutually  equiangular,  the  equal  angles 
hidng  opposite  the  equal  sides,  (Prop.  8).  The  angles  B 
and  C,  being  opposite  the  common  side  AB,  are  there^. 
fore  equal. 

Cor.  The  arc  of  a  great  circle  which  joins  the  vertex 
of  an  isosceles  spherical  triangle  with  the  middle  point  of 
the  base,  is  p  Brpendicular  to  the  base,  and  bisects  the  ver- 
tical angle  of  the  triangle ;  and,  conversely,  the  arc  of  a 


322 


SPHERICAL    GEOMETRY. 


greiit  circle  which  bisects  the  vertical  angle  of  an  isosceles 
spherical  triangle,  is  perpendicular  to,  and  bisects  the 
base. 


PROPOSITION    XI. 


If  two  angles  of  a  spherical  triangle  are  equal,  the  opposite 
sides  are  also  equal,  and  the  triangle  is  isosceles. 

In  the  spherical  triangle,  ABQ,  let  the  i_^  =  1_(7;  then 
will  the  sides,  AB  and  AC,  opposite  these  equal  angles, 
be  equal. 

For,  let  P  be  the  pole  of  the  base,  BC, 
and  draw  the  arcs  of  great  circles,  PB, 
PQ\  these  arcs  will  be  quadrants,  and  at 
right  angles  to  BQ,  (Cor.  2,  Prop.  3). 
Also,  produce  CA  and  BA  to  meet  PB 
and  P(7,  in  the  points  E  and  P.  Now, 
the  angles,  PBP  and  POP,  are  equal, 
because  the  first  is  equal  to  90°  less  the 
\_ABQ,  and  the  second  is  equal  to  90° 
less  the  equal  \_ACB\  hence,  the  A's, 
PBP  and  PQE,  are  equal  in  all  their  parts, 
since  thej  have  the  L_P  common,  the  \__PBP  =  [_PCE, 
and  the  side  PB  equal  to  the  side  PC,  (Cor.,  Prop.  9). 
PE  is  therefore  equal  to  PF,  and  [_PEC=  [__PFB. 

Taking  the  equals  PF  and  PE,  from  the  equals  PC 
and  PB,  we  have  the  remainders,  PC  and  EB,  equal; 
and,  from  180°,  taking  the  L's  PFB  and  PEC,  we  have 
the  remaining  |__'s,  AFC  and  AEB,  equal.  Hence,  the 
£^'^,AFCii\\diAEB,  have  two  angles  of  the  one  equal  to 
two  angles  of  the  other,  each  to  each,  and  the  included 
sides  equal;  the  remaining  sides  and  angles  are  therefore 
equal,  (Cor.,  Prop.  9).  Therefore,  J. (7  is  equal  to  BA^ 
and  the  A  ABC  is  isosceles. 

Cor.  An  equiangular  spherical  triangle  is  also  equilat- 
eral, and  the  converse. 


SECTION    I.  £23 

ilEMARK.  —  In  this  demonstration,  tlie  pole  of  the  base,  £6',  is  sup- 
posed to  fall  without  the  triangle,  ABC.  The  same  figure  Daay  be  used 
for  the  case  in  which  the  pole  falls  within  the  triangle ;  the  modifi- 
cation the  demonstration  then  requires  is  so  slight  and  cbvious,  that 
it  would  be  superfluous  to  suggest  it. 


PROPOSITION    XII. 

The  greater  of  two  sides  of  a  spherical  triangle  is  opposite 
the  greater  angle  ;  and,  conversely,  the  greater  of  two  angles 
of  a  spherical  triangle  is  opposite  the  greater  side. 

Let  ^-6(7  be  a  spherical  triangle,  in  which  the  angle  A 

is  greater  than  the  angle  B ;  then  is  the  side  BQ  greater 

than  the  side  AC. 

Throusrh  A  draw  the  arc  of  a 

°  \) 

great  circle,  J.i>,  making,  with  J.^,      i^   ^^r  ::- B 

the  angle  BAB  equal  to  the  angle 
ABB.  The  triangle,  BAB,  is  isos- 
celes, and  BA  =  BB,  (Prop.  11). 

In  the  A  ACD,  CD^AD>AG, 
(Prop.  1.) ;  or,  substituting  for  AD  its  equal  DB,  we  have, 
CD  -V  DB>  AC. 

If  in  the  above  inequality  we  now  substitute  CB  for 
CD+DB,  it  becomes  CB  >  CA. 

Conversely ;  if  the  side  CB  be  greater  than  the  side  GA, 
then  is  the  [_A  >  the  |_^.  For,  if  the  [_A  is  not  greater 
than  the  \_B,  it  is  either  equal  to  it,  or  less  than  it.  The 
[_A  is  not  equal  to  the  1_J5;  for  if  it  were,  the  tnanglc 
would  be  isosceles,  and  CB  would  be  equal  to  GA,  whic) 
is  contrary  to  the  hj'pothesis.  The  y_A  is  not  less  than 
the  [_B\  for  if  it  were,  the  side  (7J5  would  be  less  than  the 
side  CA,  by  the  first  part  of  the  proposition,  which  is  also 
contrary  to  the  hypothesis ;  hence,  the  \_A  must  be  greatei 
than  the  \^B. 


5J24  SP:1ERI0AL    GEOiJlETUY 

iMiOPuSlTlON    Xlll. 
Two  symmetrical  spJierical  triangles  are  equal  in  area. 

Let  ABO  and  DUF  be  two  A's  on  the  «stme  sphere, 
having  the  sides  and  angles  of  the  one  equal  to  the  sidei 
and  angles  of  the  other,  each  to 
each,  the  triangles  themselves 
Dot  admitting  of  superposition. 
It  is  to  be  proved  that  these 
^*s  have  equal  areas. 

Let  P  be  the  pole  of  a  small 
circle  passing  through  the  three 
points,  ABO,  and  connect  P 
with  each  of  the  points,  A,  B, 

and  0,  by  arcs  of  great  circles.  ISText,  through  B  draw 
the  arc  of  a  great  circle,  BP\  making  the  angle  DBF' 
•equal  to  the  angle  ABF.  Take  BF'  =  BF,  and  draw 
the  arcs  of  great  circles,  F^B,  F'F. 

The  A's,  ABF  and  BBF',  are  equal  in  all  their  parts, 
because  AB=BB,  BF^EF',  and  the  [_ABF=[_BBF', 
(Prop.  9).  Taking  from^the  [_ABO  the  \_ABF,  and 
from  the  [_BEF  the  {__BBF',  we  have  the  remaining 
angles,  FBO  and  F'EF,  equal;  and  therefore  the  A's, 
BOF  and  EFF',  are  also  equal  in  all  their  parts. 

Now,  since  the  a's,  ABF  and  DBF',  are  isosceles,  they 
will  coincide  when  applied,  as  will  also  the  A's,  BOF 
and  EFF',  for  the  same  reason.  The  polygonal  areas, 
ABOF  and  BEFF',  are  therefore  equivalent.  K  from 
the  first  we  take  the  isosceles  triangle,  FAO,  and  from  Ine 
second  the  equal  isosceles  triangle,  F'BF,  the  remainders, 
or  the  triangles  ABO  and  DBF,  will  be  equivalent. 

Remark.  —  It  is  assumed  in  this  demonstration  that  the  pole  P  falls 
without  the  triangle.  Were  it  to  fall  within,  instead  of  without,  no 
)ther  change  in  the  above  process  would  be  required  than  to  add  the 
isosceles  triangles,  PAC,  P^DF,  to  the  polygonal  areas,  to  g:«?l  thi 
Afoas  cf  the  triangles,  ABC,  DEF. 


SECTION    I. 


325 


C&r.  Two  spherical  triangles  on  tlie  sante  sphere,  or  on 
equal  spheres,  will  be  equivalent  —  1st,  when  they  are 
mutually  equilateral;  —  2d,  when  they  are  mutually  equi- 
angular;—  3d,  when  two  sides  of  the  one  are  equal  to 
two  sides  of  the  other,  each  to  each,  and  the  incluJed 
angles  are  equal;  —  4th,  when  two  angles  of  the  one  are 
equal  to  two  angles  of  the  other,  each  to  each,  and  the 
included  sides  are  equal. 


PROPOSITION   XIV. 

If  two  arcs  of  great  circles  intersect  each  other  on  the  sur* 
face  of  a  hemisphere^  the  sum  of  either  two  of  the  opposite  trir 
angles  thus  formed  will  he  equivalent  to  a  lune  whose  angle  ik 
the  corresponding  angle  formed  by  the  arcs. 

Let  the  great  circle,  AUBO,  be  the  base  of  a  hemi 
sphere,  on  the  surface  of  which  the  great  semi-circumfer 
ences,  BDA  and  CBU^  inter- 
sect each  other  at  B ;  then  will 
the  sum  of  the  opposite  tri- 
angles, BBO  and  BAU,  be 
equivalent  to  the  lune  whose 
angle  is  BBC;  and  the  sum 
of  the  opposite  triangles, 
CBA  and  BBB,  will  be  equiv- 
alent to  the  lune  whose  angle 
is  CBA. 

Produce  the  arcs,  BBA  and 
CDU,  until  they  intersect  on  the  opposite  hemisphere  at  IT; 
thezi,  since  CBU  and  BEH  are  both  semi-circumference« 
of  a  great  circle,  they  are  equal.  Taking  from  each  the 
common  part  BE^  we  have  CB  =HE.  In  the  same  way 
we  prove  BB  «=  HA,  and  AE  —  BC.  The  two  triangles, 
BBC  and  HAE^  are  therefore  mutually  equilateral,  and 
hence  they  are  equivalent,  (Prop.  13).  But  the  two  tri- 
anglen,  HAE  and  ABEy  together,  make  up  the  lune 
28 


326 


SPHERICAL    GEOMETRY. 


DEHAB\  hence  the  sum  of  the  a's,  jBi>(7and  ADE,  la 
eqiiivaleni  to  the  same  luiie. 

By  the  same  course  of  reasoning,  we  prove  that  the 
Bum  of  the  opposite  A's,  DAQ  and  DBE^  is  equi'^aleut 

the  lune  J)  CHAD ^  whose  angle  is  ABG, 


PROPOSITION    XV. 

TJu  surface  of  a  lune  is  to  the  whole  surface  of  the  sphere^ 
as  the  angle  of  the  lune  is  to  four  right  angles ;  or^  as  the  are 
which  measures  that  angle  is  to  the  circumference  of  a  great 
circle. 

JjetABFOA  he  a  lune  on  the 
mrface  of  a  sphere,  and  BCE 
an  arc  of  a  great  circle,  whose 
poles  are  A  and  F,  the  vertices 
of  the  angles  of  the  lune.  The 
arc,  BO,  will  then  measure  the 
angles  of  the  lune.  Take  any 
arc,  as  BB,  that  will  be  con- 
tained an  exact  number  of  times 
in  BO,  and  in  the  whole  circum- 
ference, BOEB,  and,  beginning  at  B,  divide  the  arc  and 
the  circumference  into  parts  equal  to  BD,  and  join  the 
points  of  division  and  the  poles,  by  arcs  of  great  circles. 
We  shall  thus  divide  the  whole  surface  of  the  sphere 
into  a  number  of  equal  lunes.  IS'ow,  if  the  arc  BO  con- 
tains the  arc  BB  m  times,  and  the  whole  circumference 
contains  this  arc  n  times,  the  surface  of  the  lune  will 
contain  m  of  these  partial  lunes,  and  the  surface  of  the 
sphere  will  contain  n  of  the  same ;  and  we  shall  have, 
Surf  lune  :  surf,  sphere  ::  m  :  n. 

But,     m  :  n  ::  BO  :  circumference  great  circle ; 
Lence,  surf,  lune  :  surf  sphere  ::  BO  :  cir.  great  circle ; 
or,         surf,  lune  :  surf,  sphere  ::  [_B0O  :  4rightangle& 


VECTION    I.  327 

This  demonstration  assumes  that  BD  is  a  common 
measuie  of  the  arc,  BC,  and  the  whole  circumference.  It 
may  happen  that  no  finite  common  measure  can  be 
found ;  but  our  reasoning  would  remain  the  same,  even 
though  this  common  measure  were  to  become  indefinitely 
small. 

Hence  the  proposition. 

Cor.  1.  Any  two  lunes  on  the  same  sphere,  or  on  equal 
spheres,  are  to  each  other  as  their  respective  angles. 

ScnoLiuM. —  Spherical  triangles,  formed  by  joining  the  pole  of  an 
arc  of  a  great  circle  with  the  extremities  of  this  arc  by  the  arcs  of 
great  circles,  are  isosceles,  and  contain  two  right  angles.  For  this 
reason  they  are  called  bi-reciangular.  If  the  base  is  also  a  quadrant, 
the  vertex  of  cither  angle  becomes  the  pole  of  the  opposite  side,  and 
each  angle  is  measured  by  its  opposite  side.  The  three  angles  are  them 
right  angles,  and  the  triaogle  is  for  this  reason  called  tri-rectangular. 
It  is  evident  that  the  surface  of  a  sphere  contains  eight  of  its  tri- 
Tectangular  triangles. 

Cor,  2.  Taking  the  right  angle  as  the  unit  of  angleSj 
and  denoting  the  angle  of  a  lune  by  A,  and  the  surface 
of  a  tri-rectangular  triangle  by  T^  we  have, 

surf,  of  lune  :  82^  ::  ^  :  4; 
whence,       surf,  of  lune  =  2A  x  T, 

Cor.  3.  A  spherical  ungula  bears  the  same  relation  to 
the  entire  sphere,  that  the  lune,  which  is  the  base  of  tne 
ungula,  bears  to  the  surface  of  the  sphere ;  and  hence, 
any  two  spherical  ungulas  in  the  same  sphere,  or  m 
equal  spheres,  are  to  each  other  as  the  angles  of  their  re- 
Ri  ective  lunes. 

PROPOSITION   XVI. 

The  area  of  a  spherical  triangle  is  measured  hy  ike  excess 
of  the  sum  of  its  angles  over  two  right  angles^  multiplied  hy 
the  tri-rectangular  triangle. 

Let  ^i^ (7  be  a  spherical  triangle,  and  DEFLK  \\\q  cir- 
cumference of  the  base  of  the  hemisphere  on  which  this 
triangle  is  situa^ted. 


S28  SPHERICAL    GEOMETRY. 

Produce  the  sides  of  the  tri- 
angle until  they  meet  this  cir- 
cumference in  the  points,  i>,  E^ 
F^  X,  K,  and  P,  thus  forming 
the  sets  of  opposite  triangles, 
DAE,  AKL ;  BEF,  BPK;  CFL, 
CDF, 

l^ow,  the  triangles  of  each  of 
these  sets  are  together  equal  to 
a  lune,  whose  angle  is  the  cor- 
responding angle  of  the  triangle,  (Prop.  14) ;  hence  we 
have, 

A  DAE  +  A  AKL  =  2^  X  I^,  (Prop.  15,  Cor.  2). 
aBEF  +  aBFK=:  2B  X  T. 
A  CFL   +  A  QDP  =  2(7  x  T. 

If  the  first  members  of  these  equations  be  added,  it  is 
evident  that  their  sum  will  exceed  the  surface  of  the 
hemisphere  by  twice  the  triangle  ABQ\  hence,  adding 
these  equations  member  to  member,  and  substituting  for 
the  first  member  of  the  result  its  value,  4iT  -\-  2  A  ABO,, 
we  have 

4T  +  2aAB0  =  2A.T~h  2B.T+  2Q.T 

or,      2T -\-    aABO=    A.T  +    B,T  +    C.T 

whence,  aABO  =    A,T  +    B.T  +    CT—IT. 

That  is,       aABC  ={A-i-  B  +  C—2)  T. 

But  A  -^  B  +  0 —  2  is  the  excess  of  the  sum  cf  the 
angles  of  the  triangle  over  two  right  angles,  and  T  de- 
notes the  area  of  a  tri-rectangulur  triangle. 

Hence  the  proposition  ;  the  area,  etc. 


SECTIOJS    1.  329 

PROPOSITION    XVII. 

The  area  of  any  spherical  'polygon  is  measured  by  the  excess 
of  the  sum  of  all  its  angles  over  two  right  angles^  taken  as 
many  times,  less  two,  as  the  polygon  has  sides,  multiplied  by 
the  tri-rect angular  triangle. 

Let  ABODE  be  a  spherical  poly- 
gon ;  then  will  its  area  be  meas- 
ured by  the  excess  of  the  sum  of 
the  angles,  A,  B,  C,  D,  and  E,  over 
two  right  angles  taken  a  number 
of  times  wliich  is  two  less  than 
the  number  of  sides,  multiplied  by 
T,  the  tri- rectangular  triangle. 
Through  the  vertex  of  any  of  the 
angles,  as  E,  and  the  vertices  of 

the  opposite  angles,  pass  arcs  of  great  circles,  thus  divi- 
ding the  polygon  into  as  many  triangles,  less  two,  as  the 
polygon  has  sides.  The  sum  of  the  angles  of  the  several 
triangles  will  be  equal  to  the  sum  of  the  angles  of  the 
polygon. 

Now,  the  area  of  each  triangle  is  measured  by  the 
excess  of  the  sum  of  its  angles  over  two  right  angles, 
multiplied  by  the  tri -rectangular  triangle.  Hence  the 
sum  of  the  areas  of  all  the  triangles,  or  the  area  of  the 
polygon,  is  measured  by  the  excess  of  the  sum  of  all  the 
angles  of  the  triangles  over  two  right  angles,  taken  as 
many  times  as  there  are  triangles,  multiplied  by  the  tri- 
rectangular  triangle.  But  there  are  as  many  triangles  as 
the  polygon  has  sides,  less  two. 

Hence  the  proposition ;  the  arex  of  any  spherical  poly- 
gon, etc. 

Ccr.  K  S  denote  the  sum  of  the  angles  of  any  spherical 

polygon,  n  the  number  of  sides,  and  T  the  tri-rectan- 

gular  triangle,  the  right  angle  being  the  unit  of  angles  j 

the  area  of  the  polygon  will  be  expressed  by 

[,S'—  2  (n  — .  2)J  X  T^  {S—2n  -f  4)  T, 
28* 


3'70  SPHERICAL    TKIGONOMETRT. 


SECTION   II 


SPHERICAL    TRIGONOMETHy. 

A  Spherical  Triangle  contains  six  parts — three  sides  and 
(Lrec  angles — any  three  of  which  being  given,  the  other 
three  may  be  determined. 

Spherical  Trigonometry  has  for  its  object  to  explain  the 
different  methods  of  computing  three  of  the  six  parts  of 
a  spherical  triangle,  when  the  other  three  are  given.  It 
may  be  divided  into  RigJd-angled  Spherical  Trigonome- 
try, and  Oblique-angled  Spherical  Trigonometry  ;  the  first 
treating  of  the  solution  of  right-angled,  and  the  second 
of  oblique-angled  spherical  triangles. 

RIGHT-ANGLED   SPHERICAL  TRIG0N0METI?3«: . 

PROPOSITION   I. 

With  the  sines  of  the  sides,  and  the  tangent  of  ONE  side 
of  any  right-angled  spherical  triangle,  two  plane  triangles  can 
Reformed  that  will  be  similar,  and  similarly  situated. 

Let  ABQ  be  a  spherical  triangle, 
right-angled  at  B ;  and  let  B  be  the 
center  of  the  sphere.  Because  the 
angle  CBA  is  a  right  angle,  the  plane 
CBB  is  perpendicular  to  the  plane 
DBA.  From  0  let  fall  CH,  perpen- 
dicular to  the  plane  DBA ;  and  us  the 


SECTION    IT.  331 

plane  CBD  is  perpendicular  to  the  plane  DBA,  CH  will 
lie  in  \\\q  plane  CBB^  and  be  perpendicular  to  the  line 
DB,  and  perpendicular  to  all  lines  that  can  be  di-awn  in 
the  plane  DBA,  from  the  point  ^(Def.  2,  B.  Vl). 

Draw  HGr  perpendicular  to  DA,  and  draw  GC\  GO 
will  lie  wholly  in  the  plane  CDA^  and  CEG  is  a  right- 
angled  triangle,  right-angled  at  H. 

We  will  now  demonstrate  that  the  angle  DGC  is  a 
right  angle. 

The  right-angled  aCBG,  gives  CH'+HG'  =  CG'   (1) 
The  right-angled  aDGH,  gives  DG'-^HG'=DH'   (2) 
13y  subtraction,     CH'  —  DG'=CG'  —  DH'        ( 3 ) 
By  transposition,  CH'  -f  DH^  =  CG'  -f  DG^       (4) 

But  the  first  member  of  equation  (4),  is  equal  to 
CD^,  because  CDH \s  a  right-angled  triangle; 

Therefore,  CD'  =  CG'  +  DG' 

Hence,  CD  is  the  hypotenuse  of  the  right-angled  tri- 
angle DGC,  (Th.  39,  B.  I). 

From  the  point  B,  draw  BF  at  right  angles  to  DA, 
and  BF  at  right  angles  to  DB,  in  the  plane  CDB  ex- 
tended ;  the  point  F  will  be  in  the  line  DC  Draw  FF, 
and  as  F  is  in  the  plane  CDA,  and  F  is  in  the  same 
plane,  the  line  FF  is  in  the  plane  CD  A.  Now  we  are  to 
prove  that  the  triangle  CHG  is  similar  to  the  triangle 
BFF,  and  similarly  situated. 

As  HG  and  BF  are  both  at  right  angles  to  DA,  they 
are  parallel ;  and  as  HC  and  BF  are  both  at  right  angles 
to  DB,  they  are  parallel ;  and  by  reason  of  the  parallels, 
the  angles  GHC  and  FBF  are  equal ;  but  GHC  is  a  right 
angle ;  therefore,  FBF  is  also  a  right  angle. 

Now,  as  Gff  and  BF  are  parallel,  and  Cff  and  BF 
are  also  parallel,  we  have, 

DB  :  DB  =  RG  :  BF 
And.  DH  I  DB  =  EC  :  BF 


S32  SPHERICAL   TRIGONOMETRY. 

Thei-efore,         EG  :  BE  =  HC  :  BF  (Th.  6,  B.  U.;, 
Or,  Ha  :  HO  =  BE  I  BE. 

Here,  then,  are  two  triangles,  having  an  angle  in  the 
one  equal  to  an  angle  in  the  other,  and  the  sides  about 
the  equal  angles  proportional;  the  two  triangles  are 
therefore  equiangular,  (Cor.  2,  Th.  17,  B.  11) ;  and  they 
are  similarly  situated,  for  their  sides  make  eqiJil  aiifjlca 
at  H  and  B  with  the  same  line,  BB, 

Hence  the  proposition. 

Scholium.  —  By  the  definition  of  sines,  cosines,  and  tanp-ents,  we 
perceive  that  CR  is  the  sine  of  the  arc  J5C,  BH  is  its  cosine,  and  RV 
its  tangent;  CG  is  the  sine  of  the  arc  AC,  and  BG  its  cosine.  Also, 
BE  is  the  sine  of  the  arc  AB,  and  BE  is  the  cosine  of  the  same  arc. 
With  this  figure  we  are  prepared  to  demonstrate  the  following  propo- 
sitions. 

PROPOSITION    II. 

In  any  right-angled  spherical  triangle,  the  sine  of  one  Me 
is  to  the  tangent  of  the  other  side,  as  radius  is  to  the  tangent 
of  the  angle  adjacent  to  the  first-mentioned  side. 

Or,  the  sine  of  one  side  is  to  the  tangent  of  the  other  side, 
as  the  cotangent  of  the  angle  adjacent  to  the  first-mentioned 
side  is  to  the  radius. 

For  the  sake  of  brevity,  we  will  represent  the  angles 
of  the  triangle  by  A,  B,  Q,  and  the  sides  or  arcs  opposite 
ho  these  angles,  by  a,  b,  c,  that  is,  a  opposite  A,  etc. 

In  the  right-angled  plane  triangle  EBE,  we  have, 
EB  :  BE  =  B  :  tSLU.BEE 

That  is,     sin.<?  :  tan.a  ==  B  :  tan.J., 
which  agrees  with  the  first  part  of  the  enunciation.     By 
reference  to  equation  (5),  Section  I,  Plane  Trigonometry, 
we  shall  find  that, 

tan.^  cot.  J.  =  i2'; 

B' 

therefore,  tan.^  = -. 

cot.  A 


SECTION  II.  833 

Substituting  tliis  value  for  tangent^,  in  tlie  preceding 
proportion,  and  dividing  the  last  couplet  by  72,  we  shall 
have, 

sin.c  :  tan.a  =  1  :  -. 

cot.A 

Or,  sin.c  :  tan. a  =  cot.^  :  B. 

Or,  -B  sin.c  =  tan.a  cot.  J.,  ( I ) 

which  answers  to  the  second  part  of  the  enunciation. 

Cor.  By  changing  the  construction,  drawing  the  tan- 
gent to  AB,  in  place  of  the  tangent  to  BC,  and  proceed- 
ing in  a  similar  manner,  we  have, 

R  sin.a  =  tan.c  cot.  0.  ( 2 ) 


jt»KOPOSITION   III. 

In  any  right-angled  spherical  triangle,  the  sine  of  the  right 
angle  is  to  the  sine  of  the  hypotenuse,  as  the  sine  of  either  of 
the  other  angles  is  to  the  sine  of  the  side  opposite  to  that  angle. 

The  sine  of  90°,  or  radius,  is  designated  by  B. 
In  the  plane  triangle,  CBG,  we  have, 

Bin.Qira  :  Ca  =^  sm.CGR  :  CH 
That  is,  B  :  sin. 5  =  sin.^  :  sih.a 

Or,  B  sin.a  =  sin.6  sin.^  ( 3 ) 

Cor.   By  a  change  in  the  construction  of  the  figure, 
drawing  a  tangent  to  AB,  etc.,  we  shall  have, 
B  :  sin.6  =  sin.(7  :  sin.c 
Or,  B sin.c  =  sin.h  sin. (7.  (4) 

Scholium. — Collecting  the  four  equations  taken  from  this  and  the 
preceding  proposition,  we  have, 

( 1 )  i?  sin.c  =  tan.a  cot.^ 

(2)  ^ sin.a  =  tan.c  cot. C 

(3)  i? sin.a  =  8in.6  sin.^ 

(4)  jR sin.c  =  8in.6  sin.c ^ 


834 


SPHERICAL    TRIGONOMETRr. 


These  equations  refer  to  the  right-angled 
triangle,  ABC;  but  the  principles  are  true 
for  any  right-angled  spherical  triangle.  Let 
us  apply  them  to  the  right-angled  triangle, 
PDC,  the  complemental  triangle  to  ABC. 

Making  this  application,  equation 

( 1 )  becomes  i?  sin.  CD  =  tan.PD  cot.  C 
(  2  )  becomes  R  sin.PD  =  tan.  CD  coLP 
( 3  )  becomes  B  sin.FD  =  Bin.FC  sin.  C 
(  4  )  becomes  B  sin.  CD  —  sin.FC  sin.P 
By  obsorving  that  sin.  CD  =~  cos. AC  =  coa.b. 

And  that  tan.FD  =  cot.  ',  t/  =  cot.^,  etc. ;  and  by  running  equa- 
tions (  »  ),  (w ),  (  0  ),  and  {p  ),  back  into  the  triangle,  ABC,  we  shall 
have, 

(5)  Bcos.b  =  cot.^  cot. 6^^ 
(  6  )  ^  cos.vl  =  cot.6    tan.c 
(7  )  R cos.A  =  cos.a  sin. C 
( 8  )  i?  cos.6  =  cos.a  cos.c 

By  observing  equation  ( 6 ),  we  find  that  the  second  member  refers 
to  sides  adjacent  to  the  angle  A.  The  same  relation  holds  in  respocl 
to  the  angle  C,  and  gives, 

(  9  )  i?  cos.c  =  cot.6  tan.a. 
Making  the  same  observations  on  ( 7  ),  we  infer, 
( 10  )  i?  cos.c  =  cos.c  sin.^. 

Observ-^tion  1.  Several  of  these  equations  can  be  de- 
duced geometrically  without  the  least  difficulty.  Foi 
example,  take  the  figure  to  Proposition  1.  The  pamllela 
ir  the  plane,  DBA,  give, 

DB  :  Dir=  BE  :  Z>(7. 

That  IS,  R  :  cos.a  =  cos.c  :  coa.b, 

A  result  identical  with  equation  ( 8  j^  and  in  words  it  is 
expressed  thus  :  Radius  is  to  cosine  of  one  side,  as  the  cosine 
of  the  other  side  is  to  the  cosine  of  the  hypotenuse. 

Observation  2.  The  equations  numbered  from  (1)  to 
(10)  cover  every  possible  case  that  can  occur  in  right-' 
angled  spherical  trigonometry;  but  the  r'ombinations  are 


SECTION    11. 


335 


too  various  to  be  remembered,  and  readily  applied  to  prac- 
tical use. 

We  can  remedy  this  inconvenience,  by  taking  the  com' 
plement  of  the  hypotenuse,  and  the  complements  of  the  two 
oblique  angles,  in  place  of  the  arcs  themselves. 

Thus,  b  is  the  hypotenuse,  and  let  b'  be  its  complement. 

Then,  5+  6'=9b°;  or,  6=  90°  — 6';  and,  sin.6  =  cos.6', 
(J0S.6  =  sin. 5';  tan. 6  =  cot.6'.  In  the  same  manner,  if  ^' 
is  the  complement  to  A, 

Then,  sin.^  =  cos.^';  cos. J.  =  sin. J.';  and,  tan.^^l  = 
cot.^';  and  similarly,  sin.C=  cos.  (7';  cos.  (7=  sin.  (7';  and 
tan.  (7=  cotC^, 

Substituting  these  values  for  5,  A,  and  C,  in  the  fore- 
going ten  equations  (a  and  c  remaining  the  same),  vre 
have. 


(12) 
(13) 
(14) 
(15) 
(16) 

(17) 
(18) 
(19) 
(2C) 


NAPIE 

Msin.c  = 
jRsin.a  = 
72  sin. a  = 
Rsin.c  = 
Rmn.b'  = 
Rsm.A'^ 
Rsm.A^  = 
Rsm.y  = 
Rsm.(7= 
Bsm.C= 


r's   circular 

:  tan. a  tan.^' 
:  tan.c  tan. (7 
:  C08.6'  cos.J.' 
=  cos.ft'  COS.  (7' 
:  tan.^'  tan.(7 
:  tan.y  tan.(? 
=  cos.a  COS.  (7' 
=  COS. a  COS. (7 
:  tan. 5'  tan. a 
:  cos.c  cos.^' 


PARTS. 

Omitting  the  consid- 
eration of  the  right  an- 
gle, there  arc  five  parts. 
Each  part  taken  as  a 
middle  part,  is  connect- 
ed to  its  adjacent  parts 
by  one  equation,  and 
to  its  extreme  parts  by 
another  equation ;  there- 
fore, ten  equations  are 
required  for  the  combi- 
nations of  all  the  parts. 


These  equations  are  very  remarkable,  because  the  first 
iiembers  a^'e  al.  composed  of  radius  into  some  sine,  and 
the  riecond  members  are  all  composed  of  the  product  of 
two  tangents,  or  two  cosines. 

To  condense  these  equations  into  words,  for  the  pur- 
pose of  assisting  the  memory,  we  will  refer  any  one  of 
them  directly  to  the  right-angled  triangle,  ABO,  in  the 
last  fi^ire. 


330  SPHERICAL    TRIGONOMETRY. 

When  the  right  angle  is  left  out  of  the  question,  a 
right-angled  triangle  consists  oi  jive  parts  —  three  sides, 
and  two  angles.  Let  any  one  of  these  parts  be  called  a 
middle  part;  then  two  other  parts  will  lie  adjacent  to  this 
part,  and  two  opposite  to  it,  that  is,  separated  from  it  by 
two  other  parts. 

For  instance,  take  equation  (H),  and  call  c  the  middle 
part;  then  A'  and  a  will  be  adjacent  parts,  and  (7'  and  6' 
opposite  parts.  Again,  take  a  as  a  middle  part ;  then  e 
and  C  will  be  adjacent  parts,  and  ^'  and  b'  will  be  oppo- 
site parts ;  and  thus  we  may  go  round  the  triangle. 

Take  any  equation  from  (H)  to  (20),  and  consider  the 
middle  part  in  the  first  member  of  tlie  equation,  and  we 
shall  find  that  it  corresponds  to  one  of  the  following  inva- 
riable and  comprehensive  rules : 

1.  The  radius  into  the  sine  of  the  middle  part  is  equal  to 
the  product  of  the  tangents  of  the  adjacent  parts. 

2.  The  radius  into  the  sine  of  the  middle  part  is  equal  to 
the  product  of  the  cosines  of  the  opposite  parts. 

These  rules  are  known  as  Napier's  Rules,  because  they 
were  first  given  by  that  distinguished  mathematician, 
who  was  also  the  inventor  of  logarithms. 

In  the  application  of  these  equations,  the  accent  maybe 
omitted  if  tan.  be  changed  to  cotan.,  sin.  to  cosin.,  etc. 
Thus,  if  equation  ( 13 )  were  to  be  employed,  it  would  be 
written,  in  the  first  instance,  i^  sin.a=  cos.^'  cos.^',  to 
insure  conformity  to  the  rule ;  then,  we  would  change  it 
into  R  sin.a  =  sin.6  sin.  J.. 

Remark. — We  caution  the  pupil  to  be  very  particular  to  take  the 
lomplements  of  the  hypotenuse,  and  the  complements  of  the  oblique 
mgles. 


SECTION    III. 


537 


SECTION   III 


OBLIQUE-ANGLED   SPHERICAL   TRIGONOMETRY. 

The  preceding  investigations  have  had  reference  to 
right-angled  spherical  trigonometry  only,  but  the  appli- 
cation of  thepe  principles  covers  oblique-angled  trigonom- 
etry also;  for,  every  oblique-angled  spherical  triangle 
may  be  considered  as  made  up  of  the  sum  or  difference 
of  two  right-angled  spherical  triangles.  With  this  ex- 
planatory remark,  we  give 


PROPOSITION   I. 

In  all  spherical  triangleSy  the  sines  of  the  sides  are  to  each 
ythery  as  the  sines  of  the  angles  opposite  to  them. 

This  was  proved  in  relation  to  right-angled  triangles  in 
Prop.  3,  Sec.  II,  and  we  now  apply  the  principle  to  ob- 
lique-angled triangles. 

Let  ABQ  be  the  triangle,  and  let 
CD  be  perpendicular  to  AB,  or  to 
AB  produced. 

Then,  by  Prop.  3,  Sec.  U,  we  have, 

JS  :  sin.  -4 C  =  sin.  A  :  sin.  CD, 
Also, 

%m,CB  :  R  =  sin.CD  :  sin.  B. 
29  w 


ilStt  SPHERICAL    TRIGONOMElKf. 

By  multiplying  these  two  proportions  together,  term 
by  term,  and  omitting  the  common  factor  M,  in  the  first 
couplet,  and  the  common  factor,  sin.6'i>,  in  the  second, 
we  have 

s'm.CB  :  sin.^C=  sin. J.  :  &m.B. 


PROPOSITION  II. 

In  any  spherical  triangle^  if  an  are  of  a  great  circle  he  lei 
fall  from  any  angle  perpendicular  to  the  opposite  side  as  a 
base,  or  to  the  base  produced,  the  cosines  of  the  other  two 
sides  will  be  to  each  other  as  the  cosines  of  the  segments  of 
the  base. 

By  the  application  of  equation  8,  (Sec.  11),  to  the  last 
Hgure,  we  have, 

B  COS. AC  =  C09, AI)  co3.I)0 

Similarly,      B  cos.BQ  =  coa.JDO  coa.BI) 

Dividing  one  of  these  equations  by  the  other,  omitting 
common  factors  in  numerators  and  denominators,  we 
have, 

COS. AC  _  COS. AD 
C08.B0       cos.BD 
Or,        COS. AC  :  cos.BO  =  cos. AD  :  cos.BJJ. 

PROPOSITION   III. 

If  from  any  angle  of  a  spherical  triangle,  a  perpendicular 
v«  let  fall  on  the  base,  or  on  the  base  produced,  the  tangents 
of  tht  segments  of  the  base  will  be  reciprocally  propcrticnal 
u>  the  cotangents  of  the  segments  of  the  angle. 

By  the  application  of  Equation  2,  (Sec.  II),  to  the  lap* 
figure,  we  have, 

B  sin.OZ)  =  t2in.ADcoi.ACJ). 


SECTION    III.  339 

Similarly,     R  sin.Ci)  =  tan.BD  cotBCD 
Therefore,  by  equality, 

tan.AD  cot ACD  =  tan.BI)  cot.BCU 
Or,        tiin,AJ)  :  tsm.BI)  =  cotBCB  :  i.otACI>, 

PROPOSITION   IV. 

2^i3  same  construction  remaining,  the  cosines  of  the  angles 
at  the  extremities  of  the  segments  of  the  base  are  to  each 
Mer  as  the  sines  of  the  segments  of  the  opposite  angle. 
Equation  7,  (Sec.  II),  applied  to  the  triangle  ACD,  gly£^^ 

R  coa.A  =  coa. CD  sin. ACD      [s] 
Also,         R  cos.B  =  coa.CD  s'm.BCD      (^ 
Dividing  equation  (■?)  by  (0,  gives 
cos.^  _  Bin. A  CD 
cos.B       Qin.BCD 
Or,  cos.^  :  cos.  J.  =  em.BCD  :  am.  A  CD, 

PROPOSITION    V. 

7^e  same  construction  remaining,  the  sines  of  the  segments 
of  the  base  are  to  each  other  as  the  cotangents  of  the  adjacent 
angles. 
Equation  1,  (Sec.  II),  applied  to  the  triangle  ACD,  givea 

R  am. AD  =  tan.OZ)  cot.^  (s) 
Sv.nilarly,  R  am.BD  ==  tan.OZ)  Qoi.B  (0 
Dividing  (a)  by  (0,  gives 

am.AD  _  cot.A 
am.BD       cot.B 
Or,  am.BD  :  i,m.AD  =  cot.-B  :  cot.^ 


340  SPHERICAL    TRIGONOMETRY. 

PROPOSITION    VI. 

The  same  construction  remaining^  the  cotangents  of  the  two 
tides  are  to  each  other  as  the  cosines  of  the  segments  of  the 
angle. 

KquatioD  9,  (Sec.  II),  applied  to  the  triangle  J.  (7i>,  givea 

B  cos.^ CD  =  cot.^ Q tan. CD  [s) 
S  railarlj,  B  cos.BOD  =  cot.BO  tan.  02)  (0 
Dividing  («)  bj  (0,  gives 

COS. A OD  _  cot. AC 
co8,BOD  ~  cotBO 
Or,  cot  AC:  cot.BO  =  coa.ACB  :  cos.BOD. 

PROPOSITION   VII. 

The  cosine  of  any  side  of  a  spherical  triangle,  is  equal  to 
the  product  of  the  cosines  of  the  other  two  sides,  plus  the 
product  of  the  sines  of  those  sides  multiplied  by  the  cosine 
of  the  included  angle. 

Let  ABQ  be  a  spherical  triangle, 
and  CB  a  perpendicular  from  the 
angle  C  to  the  side  AB,  or  to 
the  side  AB  produced.  Then,  by 
Prop.  2, 
ios.AC:co8.CB=cos.AD'.cos.BD[\) 

When   CD  falls  within  the  tri- 
angle, 

BD  =  {AB'-AD)', 

and  when  CD  falls  without  the  triangle, 
BD  =  {AD  —  AB). 

Hence,  cos.BD  =  co9.{AD  —  AB) 

Now,        cos.(^i?  -.  ^  Z>)  =  co8.{AD  —  AB), 

because  each  of  them  is  equal  to 
208. AB  COS. AD  +   sin.A^  siif.Jli),   (Eq.  10,   Prop.  :i. 
Sec.  I,  Plane  Trig.). 


SECTION    III.  341 

This  value  of  cus.  BD,  put  in  proportion  (1),  gives 
cos  ^  6' :  COS.  CB  =  cos.  J  D  :  cos.^lJ5  cos.-4Z>+sin.^i5  sin.^Z>  ( 2 ) 

Dividing  the  last  couplet  of  proportion  (2)  by  cos.^  Z>, 
observing  that 

- — Tn  =  tan.^i), 
cos.AI) 

and  we  have 

COS. J 6^  :  COS. as  =  1  :  cos.^^  +  sin.^^  tan.^D     (3) 

By  applying  equation  6,  (Sec.  II),  to  the  triangle  A  CD^ 
taking  the  radius  as  unity,  we  have 

COS.  J.  =  GOt.AC  tUTl.AB         [k) 

But,  tan.^(7cot.^C=l,(Eq.5,  Sec. I,  Plane  Trig.)     {I) 

Multiply  equation  [k]  by  inn.  AC,  observing  equation 
(0,  and  we  have 

/        tan.^Ccos.^  =  idiW.AB 

Substituting  this  value  of  tan.  J.i),  in  proportion  (3), 
we  have 
COS.  J.  C :  COS.  CB  =  1:  cos.  J.^  +  sin.  J.^  tan.  J.  C  co&.A  ( 4 ) 

Multipl^'ing  extremes  and  means,  gives 
COS.  CB=co9,.A  C  Qos.AB-\-  sin.^^  (cos.^  C  tan.-4  Q)  cos.  JL. 

But,     tan.^(7=  — ^-tt^,  or,  cos.^C  tan. J.  (7=  sin.^l(7. 

COS.^C/ 

Therefore,  cos. (75  =  cos. A (7  co^.AB -\- qih.AB  sin. AC 
cos.J.. 

If  the  sides  opposite  the  angles.  A,  B,  and  {7,  be  re^ 
dpectively  represented  by  a,  b,  and  c,  this  equation 
becomes, 

coa.a=  co8.^  cos.c  +  sin.6  sin.c  cos. A. 

This  formula  conforms  to  the  enunciation  in  respect  to 
the  side  a.  Now,  by  interchanging  h  and  a,  and  B  and  A^ 
in  the  last  equation,  we  get  the  formula  for  cos.J,  which  is, 

cos.5=cos.a  cos.c+sin.a  sin.c  cos. J?. 

29* 


842 


SPHERICAL    TRIGONOMETRY 


Interchanging  c  and  a,  and  C  and  A^  we  get  the  formula 
for  cos.c,  Avhich  is, 

cos.(?  =  cos.a  COS. 6 -f  sin. a  sin.J  cos.Ol 
Hence,  we  have  the  three  symmetrical  formulae : 
cos.a  =  C0S.5  cos.c  +  sin. ft  sin.c  cos.^^ 
cos.6  =  cos.a  cos.c  +  sin.a  sin.(?  cos.^  >     (iS') 
cos.c  =  cos.a  C08.6  +  sin.a  sin.6  cos.cj 

From  these,  by  simple  transposition  and  division,  we 
deduce  the  following  formulae  for  the  cosines  of  the 
angles  of  any  spherical  triangle,  viz : 

cos.a  —  cos.ft  cos.c^ 


cos.  J.  = 
cos.^  = 

COS.  (7  = 


sin.5  sin.c 

COS. 6  —  cos.a  cos.c 

sin. a  sin.c? 

COS.C  —  COS.«   COS. 5 


{S') 


sin. a  sin. 6 

By  means  of  these  equations  we  can  find  the  cosine  of 
any  of  the  three  angles  of  a  spherical  triangle  in  terms 
of  the  functions  of  the  sides ;  but  in  their  present  form 
they  are  not  suited  for  the  employment  of  logarithms, 
and  we  should  be  compelled  to  use  a  table  of  natural 
sines  and  cosines,  and  to  perform  tedious  numerical  ope- 
rations, to  obtain  the  value  of  the  angle. 

They  are,  however,  by  the  following  process,  tians- 
formed  into  others  well  adapted  to  the  use  of  logarithms. 

In  Eq.  34,  Sec.  I,  Plane  Trig.,  we  have 

1  +  COS.  J.  =  2cos.''JJ.. 

cos.a  —  C0S.5  C0S.6* 


Therefore,  2cos.'iJ.  =  1  + 


sin.ft  sin.tf 
(sin.J  sin.{7  —  cos. 5  cos.c)  +  cos.a 
sin.J  sin.c 


(m). 


But,  cos.(6  -\-  c)  —  coB.h  cos.c  —  sin.c  sin.6,  (Equation 
9.  Section  I,  Plane  Trig.).     By  comparing  .this  equation 


SECTION    III 


843 


with  the  secoad  member  of  equation  (wi),  we  perceive 
that  equation  ( »» )  is  readily  reduced  to 

__  cos.a  —  co8.(  b-h  c) 
Bin.6  sin.o 


2cos.«JA 


Considering  (b-hc)  as  one  are,  and  then  making  appli- 
<iation  of  equation  ( 18 ),  Plane  Trigonometry,  we  have, 


2cos.'J^  = 


2sm.  (—2—)  81I-.  (—2—) 


But, 


b  +  c 


b  +  c  +  a 


2 
b  +  c-^a 


sin.6  sin.c 

—  a;   and  if  we  put  S  to 


represent — ,  we  shall  have, 

A      sin.AS' sin.fAS' — a) 

C08.«  -  =  r-^-A ^. 

2  sin.6  sm.c 

Or,  COS.  -  =  4  /8in.^8in.(Ay— g)^ 

2        ^        sin. 6  sin.c 

The  second  member  of  this  equation  gives  the  value 
of  the  cosine  when  the  radius  is  unity.'  To  a  greater 
radius,  the  cosine  would  be  greater;  and  in  just  the  same 
proportion  as  the  radius  increases,  all  the  trigonometrical 
lines  increase ;  therefore,  to  adapt  the  above  equation  to 
our  tables  where  the  radius  is  R,  we  must  write  R  in  the 
second  member,  as  a  factor;  and  if  we  put  it  under  the 
radical  sign,  we  must  write  R^. 

For  the  other  angles  we  shall  have  precisely  similai 
equations : 

That  is,     COS.  4  =  4/^5l^^l8^ZE^' 


Sin. 6  sin.c 


COS.  ^  =  .  A'8iu.^8in.(>S-6) 


Sin. a  sin.c 


COB.  ^=  . /-K'sin.j^  6m.(S-e) 


sin. a  sin.6 


(T) 


344  SPHERICAL    TRIGONOMETRY. 

To  deduce  from  formulae  {jS),  formulae  for  the  Bines  of 
the  half  of  each  of  the  angles  of  a  spi^erical  triangle,  wc 
proceed  as  follows: 

From  Eq.  35,  Sec.  I,  Plane  Trig.,  mq  have 

2sin.'  JJ.  =  1  —  cos.^. 

Substituting  the  value  of  cos.  J.,  ''aken  from  formui« 
(aS^),  and  we  have, 

cos.a  —  cos.h  cos.<? 


2sin.'  1  J[  =  1 


sin. 6  sin.(? 


(sin.5  sin.c-f  C0S.6  cos.c)  —  cop  a      ,   . 

=  ^ T    ^    . ^- — .     [o) 

sm.6  sm.c 

But,  cos.(5  CO  c)  =  sin.6   sin.c  +  cos.J   coa.c,  (Ey^.   10, 
Sec.  I,  Plane  Trig.). 

This  equation  reduces  equation  ( » )  to 

sin.6  sin.<? 

Considering  (h  co  <?)  as  a  single  arc,  and  applying  equa- 
tion 18,  Sec.  I,  Plane  Trig.,  we  have 

2sin.  ( )  sm.  ( ) 

8m.6  sm.c 

But, = ^ c  =  S —  c,  if  we  put  S  «« 

g-f  ^  -f-g 
2        * 

Also, = h==S—b, 

Dividing  equation  [o^)  by  2,  and  making  these  substi 
tutions,  we  have 

sin.o  sm.c 
when  radius  is  unity. 


SECTION    III.  340 

When  radius  is  E,  we  have 


sin. 6  sin.c 

o     M    1       .,7.4  /i2^sin.(AS'— a)sin.(AS'--c) 

Similarly,  sm.J^  =  V -^. A ^ ^ 

*^'         -^  ^  sin. a  sin.c 

.                ^/jg-^8in.(^-a)sin.(S-^6 
And,  sm.JC  =  V ZT^TT^^TT 


sin. a  sin.6 


{IT) 


The  above  equations  are  now  adapted  to  our  tables.  We 
bIkiII  show  tlie  application  of  these  formulae,  and  those  in 
group  (T),  hereafter. 


PROPOSITION   VIII. 

The  cosine  of  any  of  the  angles  of  a  spherical  triangle,  i% 
iqual  to  the  product  of  the  sines  of  the  other  two  angles  mul- 
tiplied hy  the  cosine  of  the  included  side,  minus  the  product 
of  the  cosines  of  these  other  two  angles. 

Let  ABC  be  a  spherical  triangle,  and 
A'B'Q'  its  supplemental  or  polar  tri- 
angle, the  angles  of  the  first  being  de- 
noted by  J.,  B,  and  (7,  and  the  sides 
opposite  these  angles  by  a,  5,  (?,  respect- 
ively; A\  B',  C",  a\  6',  c',  denoting  the 
angles  and  corresponding  sides  of  the 
second. 

By  Prop.  6,  Sec.  I,  we  have  the  following  relations  be 
tween  the  sides  and  angles  of  these  two  triangles. 

A'  =  180°  —  a,  B'-=  180°  —  h,  C  =  180°  —  e; 

a'  =  180°  —  A,V  «  180°  —  B,c'=-  180°  —  C. 

The  first  of  formulift  (^),  Prop.  7,  when  applied  to  th« 
polar  triangle,  gives 

rOH   a*  rrr   POO  h*   POQ  /?'   4-   R1T1    ^'   flin.c'   COS. -4'         (1) 


346  SPHERICAL    TK  IGO  N  OME  T  R  Y. 

which,  by  substituting  the  values  of  a',  b\  c',  and  A\ 
becomes 

cos.(180°-.^)  =  cos.(180°  — J?)  cos.(180°—  C)  +  sin.(180« 
—  B)  sin.(180°  —  Q)  cos.(180°  —  a),      ( 2 ) 
But, 

cos.(180°— J.)  =  — cos.Jl,  etc.,  8in.(180°— ^)=8in.J5,etc.; 

ancl  placing  these  values  for  their  equals  in  eq.  (2),  and 

changing  the  signs  of  both  members  of  the  resulting 

equation,  we  get 

cos. J.  =  sin.^  sin.(7  cos.a  —  cos.J5  cos. (7, 

which  agrees  with  the  enunciation. 

By  treating  the  other  two  of  formulae  [S)^  Prop.  7,  in 
the  same  manner,  we  should  obtain  similar  values  for  the 
cosines  of  the  other  two  angles  of  the  triangle  ABQ\ 
or  we  may  get  them  more  easily  by  a  simple  permuta- 
tion of  the  letters  J.,  B,  (7,  a,  etc. 

Hence,  we  have  the  three  equations 
cos.  J.  =  sin.^  sin.  C 


.A  =  sin.^  sin.(7cos.a  —  cos.^  cos.C") 

.B  =  sin. J.  sin.C^cos.i  —  cos. J.  cos. (7  V     (^) 

.0  =  sin. J.  sin.i?  cos.f  —  cos. J.  co^.B) 


cos 
cos 

By  transposition  and  division,  these  equations  become 

^^^  cos.-A -f  cos.^  COS. (7    ,o\ 

sm.^  sin.  (7 
T       QOQ.B  +  cos.^  COS.  (7 

COS.6  =  . T—-— 77 

sin.^  sm.C^ 
COS.  (7  +  COS.  A  C0S.5 

COS.C  =  r-— i r  — ^ 

Sin.  J.  sm.^ 

From  these  we  can  find  formulae  to  express  the  sine  or 
tL^  cosii.t)  of  one  half  of  the  side  of  a  spherical  triangle, 
m  terms  of  the  functions  of  its  angles ;  thus : 

Add  1  to  each  member  of  eq.  (3),  and  we  have 

COS.  J.  4-  C0S.J5  cos.  (7  -f  sin.^  sin.  (7 


1  4  cos.a  = 


sin.^  sin.  (7 


SECTION    .II.  847 

COS.^  +  C0S.(j5  —  O) 


%\\i.B  sin.  6^ 
But,  1  +  cos.a  =  2cos.'  Ja ;  heuce, 

2cos.'  \a  = ^. — ^— .^^— 


Qin.B  sin.  (7 

nnd  Binje  cos.A  +  cos.(jB  —  C)  =  2cos.J(J.  +  J5—  (7)co8.} 
{A\-Q—B)  (Eq.l7,  Sec.  I,  Plane  Trig.),  we  have 

2co8.'  U  =  2C08.KA  +  ^  -  (7)co8.K^  +  (7-  ^) 

sin.^  sin.  (7 

Make  ^  +  ^  +  (7=  2/S';  then  A  +  ^—  (7=  2S—20, 
A  +  Q—B^  2S—2B,  i{A-\-B—a)  =  S—  0,  and  i(A 
-I  g—B)  =  S—B;  whence 

sm.jB  sin.  (7 

,  /^^i:(^=0)co8.(AS'— J5) 
'  ^  ^  sin.jBsin.(7 

o-      -1      1  11  \    / G0S.(S—A)C08,(S—C)   V     /TT/v 

Similarly,  cos.JS  =  \/ ^^-^^->  >   (F') 

and,  COS.J.  =  \/2£iI?^oi;^ 

^  Sin. A  sin.J5 

To  find  the  sin.Ja  in  terms  of  the  ftinctions  of  tlie 
angles,  we  must  subtract  each  member  of  eq.  ( 3 )  from  1, 
by  which  we  get 

^      cos.^  +  COS.jS  cos.(7 
sin.^  sin.C? 

But,  1 — cos.a=  2sin.'Ja;  hence  we  have, 

p  .    ,-        (sin.j5  sin.  (7 — cos.^  cos. (7) — cos.  J. 

^Sin*    *fl  —  — — z =r ; — —  • 

sm.jB  sin.O' 

Operating  upon  this  in  a  manner  analogous  to  that  l»j 
which  cos.Ja  was  found,  we  get, 


348  SPHERICAL   TRIGONOMETRY. 

^        \        sm.B  sin.  (7        / 

sm. J5  =  { ^.        ._  V     — ;  \2       )    ( IT) 

(         sin.^  sin.  (7        J  /     ^     ' 

Bin.i.-  f-cos.^co3.(^-{7)U 


L.Jc.=  { 


sin.^  sin.^         J 


L  the  first  equation  in  (W)  be  divided  by  the  first  m 
{  ^'^ ),  we  shall  have, 

And  corresponding  expressions  may  be  obtained  fof 
tan. J 6  and  tan.J^?. 


NAPIER'S   ANALOGIES. 

If  the  value  of  cos.^?,  expressed  in  the  third  equation 
of  group  (aS'),  Prop.  7,  be  substituted  for  cos.<?,  in  the 
second  member  of  the  first  equation  of  the  same  group, 
we  have, 

COS. a  =  cos.a  cos.'&  -|-  sin. a  sin. 6  cos. J  cos.  C+  sin.J  sin.c  cos.^; 

which,  by  writing  for  cos.^'J  its  equal,  1  —  sin.'ft,  becomes, 

coa.a=cos.a — cos.a  sin.^i-fsin.a  sin. 6  cos.&  cos.  C-fsin.ft  sin.c  cos.^. 

Or,  0  =  — cos.a  sin.^5-f-sin.a  sin.6  cos.6  cos.C+sin.i  sin.c  cos.J.. 

Dividing  through  by  sin. 5,  and  transposing,  we  find 

cos.^  sin. (j=  cos.a  sin.6 — sin.a  cos.5  cos. (7; 

-      cos.^  sin.5  —  sin.a  cos. 5  COS. (7     ,,, 

.once,  cos.^l  = ; (1) 

sm.(? 

By  substituting  the  value  of  cos   .  in  the  second  of  the 

equations  of  group  (S),  Prop.  7 ;  or,  merely  writing  £  Ibi 

A,  and   interchanging  h  and  a,  in  the  above  value,  foi 

sK)S.  A,  we  obtain, 

^     C0S.5  sin.a— sin.5  cos.<*  cos.<7.     ,,     • 
cos.^  = ; (2) 


SECTION    III.  349 

Adding  equations  (1)  and  (2),  member  to  member, 
we  have, 

^     8in.(a+5)— sin.fa-l-J)  cos.C. 

COS.  J.  +  cos.-B  = i^ '— , 

sm.c 

by  reraembtring  that  sin.a  cos.ft  +  cos.a  sin.6«Bin.(a+6). 
(See  Eq.  (7 ),  Sec.  I,  Plane  Trig.). 

Wh  3nce,  cos. J.  +  cos.jB  =  (1  —  cos.  C)  ?^^^  — \     ( 3 ) 

In  any  splierical  triangle  we  have,  (Prop.  I), 

sin.-4.  :  sin.  J5  : :  sin.a  :  sin. 6 ; 

And  therefore,  sin.^  +  sin.^  :  sin.-B  : :  sin.a  +  sin.5  : 
sin.  6. 

„  .      ^   .    .     T>     (sin.a  +  sin. 5)  sin.J? 

Hence,  sm.^  +  sm.jS  =  ^^ . — ^-^ . 

sm.6 

But,  ?i5:^  =  «4?4  which  value  of  ?^4. 5n  the  abore 

sin. 6       sin.tf  sm.o 

equation,  gives 

.   .    .     ^     (sin.a  +  sin.5)  sin.(7         ,  .. 

sm.  J.  +  sm.-B  =  ^ -, '- .        (4) 

sm.c 

Dividing  equation  (4)  by  equation  (3),  member  by 

member,  we  obtain, 

sin.  J.  +  sin.5  __     sin.  (7        sin.a +  8in.6       .^. 

cos.JL  +  cos.^      1 — cos.(7       sin.(a4-J) 

Comparing  this  equation  with  Equations  (20)  and  (2G), 
dec.  I,  Plane  Trigonometry,  we  see  that  it  can  bo  re- 
duced to 

-,.._,.  .  ^ />     sin.a  +  sin. 6         ,«v 

tan.J(^  +  ^)='Cot.J(7x     .    ,    .  ,,  (6) 

^  sm.(a  +  6) 

Again,  from  the  proportion, 

sin.  J.  :  sin.J5  : :  sin.a  :  sin.5, 

we  likewise  have, 

ein.^  —  sin.^  :  sin.^  ::  «in.a — sin.J  :  ain.i; 

80 


350  SPHERICAL    TRIGONOMETRY. 

neuce,   sm.^  —  sm.^  =  (sm.a  —  sin.o) =  (sm.a  — 

sin.6 

8111.  o) . 

sm.c 

Dividing  this  equation  by  equation  (3),  member  bj 

niember,  we  obtain, 

sin.^  —  sin.^  _     sin.  (7         sin. a — sin. 5 

cos.^  +  cos.^      1 — 008.(7       sin.(a4-6)* 

Comparing  this  with  Equations  (22)  and  (26)^  Sec.  I, 

Plfcue  Trigonometry,  we  see  that  it  will  reduce  to 

1/^        T^i  .  , /-/      sin.a — sin. 6         _. 

tan.|(^  —  ^)  =  cot.i(7  x  — .— t — -j^.       (7 
^^  '  ^  sin. (a  -f  h) 

Now,sin.a  +  sin.6  =  2sin.(-2~)cos.(— ^^— );  Eq.  (15), 
Sec.  I,  Plane  Trig.). 

and,  sin.  (a  +  5)  =  2sin.("2— )  c<>s.(^^);  Eq.  (30), 

Sec.  I,  Plane  Trig.). 

Dividing  the  first  of  these  by  the  second,  we  have 

a  —  h^ 
sin. a  +  sin. 6 


'^°^-(-2-) 

sin.(a  4-6)  /«  +  h^ 

^  '        cos.' 


Writing  the  second  member  of  this  equation  for  ita 
&rst  member  in  Eq  (6),  that  equation  becomes 

tan.  \{A  +  B\  =  cot.  K^^^'  ^ TS'     ^^^ 
^  COS.  J(a-f  6) 

Bj  a  similar  operation,  Eq.  (7)  may^be  reduced  to 

tan.  i{A  -  .g)  =  cot.  |(7'!"-  ^.'^"fl     (9) 
^^  y  ^     sin.i(a+^') 

Equations  ( 8 )  and  ( 9 )  may  be  resolved  into  the  pro- 
portions 
cos.  J(a  +  h)  :  cos.  J(a  —  5)  : :  cot.  J  (7  :  tan.  J(JL  +  jB)  ; 
sin.  J(a  +  5)  :  sin.  J(a  —  5)  : :  cot.  J (7  :  tan.  J(J.  —  B). 
These  proportions  are  known  as  Napier's  1st  and  2d 


SECTION    III.  351 

Analogies,  and  may  be  advantageously  used  in  the  solu- 
tion of  spherical  triangles,  when  two  sides  and  the  in- 
cluded angle  are  given. 

When  expressed  in  language,  these  proportions  fur- 
nish the  following  rules: 

1.  The  cosine  of  the  half  sum  of  any  two  sides  of  a  spheri- 
cal triangle  is  to  the  cosine  of  the  half  difference  of  the  same, 
tides,  as  the  cotangent  of  half  the  included  angle  is  to  *he 
tangent  of  the  half  sum  of  the  other  two  angles. 

2.  The  sine  of  the  half  sum  of  any  two  sides  of  a  spheri- 
cal triangle  is  to  the  sine  of  the  half  difference  of  the  same 
tides,  as  the  cotangent  of  half  the  included  angle  is  to  the 
tangent  of  the  half  difference  of  the  other  two  angles. 

The  half  sum,  and  the  half  difference  of  two  angles 
of  a  spherical  triangle,  may  be  found  by  these  rules,  w^hen 
two  sides  and  the  included  angle  ;irc  given;  and  by  add- 
ing the  hrlf  sum  to  the  half  difference,  we  get  the 
greater  of  these  two  angles,  and  by  subtracting  the  half 
difference  from  the  half  sum,  we  get  the  smaller.  The 
third  side  may  then  be  found  by  proportion. 

We  have  analogous  proportions  applicable  to  the  case 
in  which  two  angles  and  the  included  side  of  a  spherical 
triangle  are  given. 

To  deduce  these,  let  us  represent  the  angles  of  the  tri- 
angle ]»y  A,  B,  and  Q,  and  the  opposite  sides  by  a,  h,  and 
c ;  A',  B',  (7',  a',  h',  c',  denoting  the  corresponding  angles 
and  sides  of  the  polar  triangle. 

Kow,  Eq.  (9)  is  applicable  to  any  spherical  triangle, 
and  when  applied  to  the  polar  triangle,  it  becomes 

But  by  Prop.  6,  Sec.  I,   Spherical  Geometry,  we  have 

A'  =  ISO'^  —  a,B'=-  180°  —  h,C'  =  180°  —  c, 

a'  =  180°  —A,y  =  180°  —B,c'  =  180°  —  C, 

Whence,  l{A'-B')=^\{h-a\  l{a'  +  h')  =  180°--4_|  J^^ 

j(a'  -  5')  -  \(B  -  ^),  JC^  =  90-  -  \c. 


852  SPHERICAL    TRIGONOMETRY. 

By  the  substitution  of  these -values  in  Eq    (w),  th>it 

equation  l)ecomes 

1/7         X      sin.  i(5  —  A)  ^ 
tan.  Vb  —  a)  =  -. — f;  .    .    ^!  tan.  ic, 
^'  ^      sin.  J(J.  +  jB)  ^' 

or,        tan.  i(a  —  5)  =  ^^^'  ?L  T  J  ^^^'  J^>         ^^^ 
'  ^^  ^      sin.  i(J.  +  ^)  ^  ' 

Bince  tan.  J(6  —  a)  =  —  tan.  J(a  —  5),  and  sin.  i(J5  —--4)  = 
—  sin.  i{A  —  ^). 

By  applying  Eq.  (8)  to  the  polar  triangle,  and  treating 
the  resulting  equation  in  a  manner  similar  to  the  above, 
we  find 

tan.  i{a  +  5)  =  ^Q«' K^  ~  g  tan.  },,  (q) 

cos.  ^{A  -\-  B)  ^  ' 

Equations  {p)  and  (?)  may  be  resolved  into  the  fol- 
lowing proportions. 

sin.  J(J.  +  B)  :  sin.  J(J.  —  5)  : :  tan.  ^o  :  tan.  J(a  —  h): 
COS.  J(J.  +  B)  :  COS.  J(J.  —  B)  ::  tan.  Jc  :  tan.  J(a  H   6). 

These  proportions  are  called  Napier's  3d  and  4th 
Analogies,  and  when  expressed  in  words  become  the  fol- 
lowing rules: 

1.  The  cosine  of  the  half  sum  of  any  two  angles  of  a 
spherical  triangle  is  to  the  cosine  of  the  half  difference  of  the 
same  angles^  as  the  tangent  of  half  the  included  side  is  to  the 
tangent  of  the  half  sum  of  the  other  two  sides. 

2.  The  sine  of  the  half  sum  of  any  two  angles  of  a  spheri- 
cal triangle  is  to  the  sine  of  the  half  difference  of  the  sami 
angles,  as  the  tangent  of  half  the  included  side  is  to  the  tan- 
gent of  the  half  difference  of  the  other  two  sides. 

The  half  sum,  and  the  half  difference  of  two  sides  of 
tt  spherical  triangle,  may  be  found  by  these  rules,  when 
two  angles  and  the  included  side  are  given ;  and  by  add- 
ing the  half  sura  to  the  half  difference,  we  get  the  greater 
>f  these  sides,  and  by  subtracting  the  half  difference 
from  the  half  sum,  we  get  the  smaller. 


SECTION   IV.  868 


SECTION  IV. 


ISPHEllICAL  TRIGONOMETRY  APPLIED. 
SOLUTION  OF  RIGHT-ANGLED  SPHERICAL  TRIANGLES. 

A  GOOD  general  conception  of  the  sphere  is  essential 
to  a  practical  knowledge  of  spherical  trigonometry,  and 
this  conception  is  hest  obtained  by  the  examination  of 
an  artificial  globe.  By  tracing  out  upon  its  surface  the 
various  forms  of  right-angled  and  oblique-angled  tri- 
angles, and  viewing  them  from  different  points,  we  may 
soon  acquire  the  power  of  making  a  natural  representa- 
tion of  them  on  paper,  which  will  be  found  of  much  as- 
sistance in  the  solution  and  interpretation  of  problems. 

For  instance,  suppose  one  side  of  a  right-angled 
spherical  triangle  to  be  56°,  and  the  angle  between  this 
side  and  the  hypotenuse  to  be  24°.  What  is  the  hypote- 
nuse, and  what  the  other  side  and  angle  ? 

A  person  might  solve  this  problem  by  the  application 
of  the  proper  ef|uations  or  proportions,  without  really 
comprehending  it ;  that  is,  without  being  able  to  form  a 
distinct  notion  of  the  shape  of  the  triangle,  and  of  its 
relation  to  the  surface  of  the  sphere  on  which  it  is 
situated. 

If  we  refer  this  triangle  to  the  common  geographical 
globe,  the  side  66°  may  be  laid  off  on  the  equator,  or  on 
a  meridian.  In  the  first  case,  the  hypotenuse  will  be  the 
arc  of  a  great  circle  drawn  through  one  extremity  of  the 
Bide  56°,  above  or  below  the  equator,  and  making  with 
80*  X 


PM 


SPHERICAL    TKIGOyOMETRY. 


it  an  angle  of  24° ;  the  other  side  will  be  an  arc  of  a 
meridian.  In  the  second  case,  the  side  56°  falling  on  a 
meridian,  the  hypotenuse  will  be  the  arc  of  a  great  circle 
drawn  through  one  extremity  of  this  side,  on  the  right 
or  left  of  the  meridian,  and  making  with  it  an  angle  of 
24° ;  the  other  side  will  be  the  arc  of  a  great  circle,  at 
Tight  angles  to  the  meridian  in  which  the  given  side  lies. 

Generally  speaking,  th3  apparent  form  of  a  spherical 
triangle,  and  consequently  the  manner  of  representing 
it  on  paper,  will  difler  with  the  position  assumed  for  tl  e 
eye  in  viewing  it.  From  whatever  point  we  look  at  a 
sphere,  its  outline  is  a  perfect  circle  in  the  axis  of  which 
the  eye  is  situated;  and  when  the  eye  is,  as  will  be  here- 
after supposed,  at  an  infinite  distance,  this  circle  will  be 
a  great  circle  of  the  sphere.  All  great  circles  of  the 
sphere  whose  planes  pass  through  the  eye,  will  seem  to 
be  diameters  of  the  circle  which  represents  the  outline 
of  the  sphere. 

We  will  now  suppose  the  eye  to  be  in  the  plane  of  the 
equator,  and  proceed  to  construct  our  triangle  on  paper. 

Let  the  great  circle, 
PASA^y  represent  the  out- 
line of  the  sphere,  the  di- 
ameter AA^  the  equator, 
and  the  diameter  P^S'  the 
central  meridian,  or  the 
m'^-ridian  in  whose  plane 
the  eye  is  situated.  Let 
AB—  56°,  represent  the 
given  side,  and^  6^,making 
with  AB  the  angle  BA  (7= 
24°,  the  hypotenuse,  then  will  BO,  the  arc  of  a  meridian, 
be  the  other  side  at  right  angles  to  AB,  and  the  triangle, 
ABC,  corresponds  in  all  respects  to  the  given  triangle. 

Again  measure  off  56°  from  P  to  Q,  draw  the  arc  DQ, 
make  the  arc  A'G  equal  to  24°,  and  draw  the  quadrant 
PEG.  The  triangle  FQP  will  also  reoresent  the  given 
triangle  in  every  particular. 


SECTION   IV.  355 

"We  know  from  the  construction,  that  DV,  =  24°,  is 
greater  than  BO,  and  that  AOis  greater  than  AB,  that 
is,  greater  than  56°. 

In  like  manner,  we  know  that  >4',  =  24°,  is  greater 
than  QRj  and  that  FR  is  greater  than  FQ,  because  FB 
18  more  nearly  equal  toFG,  =90^,  thanP  Q  is  to  FA,  =90° 

For  illustration  and  explanation,  we  also  give  the  fol- 
lowing example: 

In  a  light-angled  spherical  triangle,  there  are  given, 
the  hypotenuse  equal  to  150°  33'  20",  the  angle  at  the 
base,  23°  27'  29",  to  find  the  base  and  the  perpendicular. 
Let  A'BQ  in  the  last  figure,  represent  the  triangle  in 
which  A'Q=  150°  33'  20",  the  [__  BA'Q^  23°  27'  29", 
and  the  sides  A'B  and  BG  are  required. 

This  problem  presents  a  right-angled  spherical  tri- 
angle, whose  base  and  hypotenuse  are  each  greater  than 
90° ;  and  in  cases  of  this  kind,  let  the  pupil  observe, 
that  the  base  is  greater  than  the  hypotenuse,  and  the  oblique 
angle  opposite  the  base,  is  greater  than  a  right  angle.  In 
all  cases,  a  spherical  triangle  audits  supplemental  triangle 
make  a  lune.  It  is  180°  from  one  pole  to  its  opposite, 
whatever  great  circle  be  traversed.  It  is  180°  along  the 
equator  ABA',  and  also  180°  along  the  ecliptic  ACA^, 
The  lune  always  gives  two  triangles;  and  when  the 
sides  of  one  of  them  are  greater  than  90°,  we  take  the 
triangle  having  supplemental  sides ;  hence  in  this  case 
we  operate  on  the  triangle  ABQ, 

AC  m  greater  than  AB,  therefore  A*B  is  greater  than 
the  hypotenuse  AO, 

The  [_AQB  is  less  than  90°;  therefore,  the  adjacent 
angle  A'CB  is  greater  than  90°,  the  two  together  being 
equal  to  two  right  angles. 

These  facts  are  technically  expressed,  by  saying,  that 
the  sides  and  opposite  angles  are  of  the  same  affection,* 

*  Same  affection :  that  is,  both  greater  or  both  less  than  90°.  Dif 
/'rent  affection  :  the  one  greater,  the  other  less  than  90°. 


8i)6  SPHERICAL    TRIGONOMETRY. 

Now,  if  the  two  sides  of  a  right-angled  spherical  triaj^.^le 
are  of  the  same  affection,  the  hypotenuse  will  be  less  than 
90° ;  and  if  of  different  affection^  the  hypotenuse  will  be 
greater  than  90"^. 

If,  in  every  instance,  we  make  a  natural  construction 
of  the  figure,  and  use  common  judgment,  it  will  b<i  im- 
possible to  doubt  whether  an  arc  must  be  taken  gr^atet 
or  less  than  90°. 

We  will  now  solve  the  triangle  ACB,  AO^  180®—- 
150°  33'  W  =  29°  26'  40". 

To  find  BO,  we  use  Eq.  (3)  or  (13),  Prop.  3,  Sec.  IT., 

thus: 

5,  sin.  29°  26'  40"     .      9.691594 

A,  sin.  23°  21'  29"     .       9.599984 
a,sin.  11*^  17'     7"     .       9.291578 

To  find  AB,  we  use  equation  ( 1 )  or  ( H ),  thus : 

a,  tan.  11°  17'     7"     .       9.300016 
,     A,  cot.  23°  27'  29"     .     10.362674 

c,sin.  27°  22'  32"     .       9.662690 

180 

^'^=152°  37'  28" 


PRACTICAL  PROBLEMS  IN  RIGHT-ANGLED    SPHERICAL 
TRIGONOMETRY. 

1.  In  the   right-angled  spherical  ^^ 
triangle  ABQ,  given  AB  =  118°  21'     a^.- 

4",  and  the  angle  A  =  23°  40'  12",  to        ^^""--.^,^7 
find  the  other  parts.  B 

.       { AQ,  116°  17'  56"',  the  angle  (7,  100°  59'  26"; 
'  I      and  BC,  21°  5'  42". 

2.  In  the  right-angled  spherical  triangle  ABC,  given 
AB  53°  14'  20",  and  the  angle  A  91°  25'  53",  to  find 
the  other  parts. 

.       r  J.(7,  91°  4'  9";  the  angle  (7,  53°  15'  8"; 
^"*'l     and  J5 (7,  91°  47' 10' . 


SECTION  IV.  357 

8.  In  the  right-angled  spherical  triangle  ABC^  given 
XB  102°  50'  25'',  and  the  angle  A  113°  14'  37",  to 
find  the  other  parts. 

.      [AC,  84°  51'  36";  the  angle  (7,  101°  46'    56"; 
^^'l      and^C,  113°  46' 27". 

4.  In  the  right-angled  spherical  triangle  ABQ,  given 
A  B  48°  24'  16",  and  BQ  59°  38'  27",  to  find  the 
other  parts. 

.       f  AC,  70°  23'  42";  the  angle  A,  m""  20'  40^'; 
\      and  the  angle  (7,  52°  32'  56". 

5.  In  the  right-angled  spherical  triangle  ABC,  given 
AB  151°  23'  9",  and  BC  16°  35'  U",  to  find  the 
other  parts. 

.       (AC,  147°  16'  51";  the  angle  (7,  117°  37'  25"; 
^**  t      and  the  angle  A,  31°  52'  49". 

6.  In  the  right-angled  spherical  triangle  ABC,  given 
AB  73°  4'  31",  and  ^C86°  12'  15,"  to  find  the  other 
parts. 

.       f  ^(7,  76°  51'  20";  the  angle  A,  77°  24'  23"; 
I      and  the  angle  C,  73°  29'  40". 

7.  In  the  right-angled  spherical  triangle  ABC,  given 
AC  118°  32'  12",  and  AB  47°  26'  35",  to  find  the 
other  parts. 

.       r  5(7,  134°  56'  20";  the  angle  A,  126°  19'  2"; 
I      and  the  angle  C,  56°  58'  44". 

8.  In  the  right-angled  spherical  triangle  ABC,  given 
AB  40°  18'  23",  and  AC  100°  3'  7",  to  find  the 
olher  parts. 

.       f  The  angle  A,  98°  88'  58" ;   the   angle 
I      (7,  41°  4'  6" ;  and  5(7,  103°  13'  52". 
y.  In  the  right-angled  spherical  triangle  ABC,  given 
AC  61°  3'  22",  and  the  angle  A  49°  28'  12",  to  find 
the  other  parts. 

?  ^5,  49°  36'  6" ;  the  angle  (7,  60°  29'  20'' ; 
^**\      and  5(7,  41°  41' 32". 
10.  In  the  right-angled  spherical  triangle  ABC^  given 


SPHERICAL    TRl  JONOMETRr. 

AB  29°  12'  50",  and  tlie  angle  0  37°  26'  21",  to  lind 
the  other  parts. 

/-Ambiguous;  the  angle  A,  Qb""  27'  57",  or  its 
Ans.'l      supplement;  AC,  53°    24'  13",  or  its  sup- 

l     plement;  BO,  46°  55^  2",  or  its  supplement 

11.  In  the  right-angled  spherical  triangle  ABO,  given 
AB  100°  10'  3",  and  the  angle  0  90°  14'  20",  to  find 
the  other  parts. 

rAO,  100°  9'  52",  or  its  supplement;  BO, 
Ans.<      1°  19'  55'',  or  its  supplement;  and  the 
L     angle  ^,1°  21'  12",  or  its  supplement. 

12.  In  the  right-angled  spherical  triangle  ABO,  given 
AB  54°  21'  35",  and  the  angle  0  61°  2'  15",  to  find 
the  other  parts. 

rBO,  129°  28'  28",  or  its  supplement;   AC, 

An8.<      111°  44'  34",  or  its  supplement;  and  the 

I     angle  A,  123°  47'  44",  or  its  supplement. 

13.  In  the  right-angled  spherical  triangle  ABO,  given 
AB  121°  26'  25",  and  the  angle  0  111°  14'  37",  to 
find  the  other  parts. 

rThe  angle  A,  136°  0'  5",  or  its  supplement; 
AnsA      AO,  m""  15'  38",  or  its  supplement;  ard 
L     BO,  140°  30'  57",  or  its  supplement. 


QUADRANTAL    TRIANGLES. 

The  solution  of  right-angled  spheri- 
ral  triangles  includes,  also,  the  solu- 
noi!  of  quadrantal  triangles,  as  may  be 
seeij  by  inspecting  the  adjoining  fig- 
ure When  we  have  one  quadrantal 
triangle,  we  have  four,  which  with  one 
right-angled  triangle,  fill  up.the  whole  hemisphere^ 

To  eflfect  the  solution  of  either  of  the  four  quadrantal 
mangles,  APO,  AP'O,  A'PO,  or  J.'P'(7,  it  is  sufficient 
to  solve  the  small  right-angled  spherical  triangle  ABO, 


SECTION    IV.  359 

To  the  half  lune  AF'B,  we  add  the  triangle  ABC, 
and  we  have  the  quadrantal  triangle  AF'C;  and  by  sub- 
tracting the  same  from  the  equal  half  lune  AFB,  we 
have  the  quadrantal  triangle  PA  0, 

When  we  have  the  side,  AC,  of  the  same  triangle,  we 
have  its  supplement,  A'C,  which  is  a  side  of  the  triangles 
A'FC,  and  A'F'O.  When  we  have  the  side,  CB,  of 
the  small  triangle,  by  adding  it  to  90°,  we  have  P'C',  a 
side  of  the  triangle  A'P'C\  and  subtracting  it  from  90*^, 
we  have  P(7,  a  side  of  the  triangles  APC,  and  AlPC, 

PROBLEM    I. 

In  a  quadrantal  triangle,  there  are  given  the  quadrantal 
gidej  90°,  a  side  adjacent,  42°  21',  and  the  angle  opposite 
this  last  sidey  equal  to  36°  31'.     Required  the  other  parts. 

By  this  enunciation  we  cannot  decide  whether  the  triangle  APQ 
or  AP'Gf  is  the  one  required,  for  AC  =  42°  21'  belongs  equally 
to  both  triangles.     The  angle  AFC  =  AP'C^  36°  31'  =  AB 

We  operate  wholly  on  the  triangle  ABC. 

To  find  the  angle  A,  call  it  the  middle  part. 
Then,  R  cos.  CAB  =  R  &mPA  C  =  cot.  J.  C  tan.^^. 

cot.^C     =     42°  21'        .         10.040231 
tan.J^^     ==     36°  31'         .  9.869473 

cos.  CAB  =     35°  40'  51"  9.909704 

"       90° 

PAC  =     54°  19'    9" 
PAC=  125° 40' 51" 

To  find  the  angle  (7,  call  it  the  middle  part 
R  coi.  A CB  =  sin.  CAB  cos.  AB, 


Bin.CAB  =     35°  40' 51" 
cos.^^     =     36°  31' 

9.765869 
9.905085 

COB. A  CB  =     62°    2' 45" 

180° 

9.670964 

ACP  «   A'CP  =  117°  57' 15" 

860 


SPHERtCAL    TRIGONOMETRY. 


To  find  the  side  BO,  call  it  the  middle  part. 
Rsin.BO  =  tan.^j5  cot.^C^. 


tan.^^     ==     36°  31'    0" 
cot  ACB  =     62°    2' 45" 

9.869473 
9.724835 

Bin.BC     =     23°    8' 11" 
90° 

9.594308 

PC  =     66°  51'  49" 
PC  =  113°    8'  11" 

We  now  have  all  the  sides,  and  all  the  angles  of  the 
four  triangles  in  question. 

PROBLEM  IT. 

In  a  quadrantal  spherical  triangle ,  having  given  the  qiuid- 
rantal  side,  90°,  an  adjacent  side,  115°  09',  and  the  included 
angle,  115°  55',  to  find  the  other  parts. 

This  enunciation  clearly  points  out 
the  particular  triangle  A'P'O,  A'P' 
=  90°  ;  and  conceive  A'0=  115°  09'. 
Then  the  angle  F'A'C=  115°  55'  = 
P'D, 

From  the  angle  P'^'^take  90°,  or 
P'A'B,  and  the  remainder  is  the  angle 
OA'B^  BAO^  25°  55'. 

We  here  again  operate  on  the  triangle  ABQ,  A'C, 
taken  from  180°,  gives 

64°  51'  ^  AC. 

To  find  EC,  we  call  it  the  middle  part. 

R  sin.  5(7=  sin.^Csin.^^C. 

Bm.AC     =    64°  51'  .        9.956744 

Bin.BAG  =     25°  55'  .        9.640544 


8in.^C     =     23°  18'  19" 

90° 

PC  «  113°  18'  19" 


9.597288 


SECTION    IV.  861 

To  find  AB,  we  call  it  the  middle  part. 

R^m.AB  =  tan.BCcoi.BAC. 

tau.BC     =     23°  18' 19"     .         y.634251 

?ot.BAC  =     25°  55'  .         9.313423 


s'mAB      =     G2°  2G'    8"     . 

8.947674 

180° 

AB  =  117°  33'  52"  = 

the  angle  AP'a 

To  find  the  angle  (7,  we  call  it  the  middle  part. 

R  COS.  C  =  QotA  C  tan.BC. 

cot.AC    =     04°  51' 

9.671634 

taii.^C    =     23°  18'  19"      . 

9.634251 

eos.C       =     78° 

9.305885 

18U°  19'  53"      . 

P'CA  =  101°  40'    7" 

Thus  we  have  found  the  side  rC  =   113°  18'  19" 

The  angle  AI^C  =   117°  33'  52" 

"      P'CA'  =   101°  40'    7" 


1 


PRACTICAL    PROBLEMS. 


1.  In  a  qnadrantal  triangle,  given  the  qnadrantal  side, 
)0®,  a  side  adjacent,  67°  3',  and  the  included  angle,  49** 
18',  to  find  the  other  parts. 

rThe  remaining  side  is  53°  5' 44" ;  the  angle 
Ans.   <      opposite  the  qnadrantal  side,  108°  32'  29"; 
I      and  the  remaining  angle,  60°  48'  54". 

2.  In  a  qnadrantal  triangle,  given  the  qnadrantal  side, 
90°,  one  angle  adjacent,  118°  40'  36",  and  the  side  op- 
posite this  last-mentioned  angle,  113° 2'  28",  to  find  the 
other  p.arts. 

f  The  remaining  side  is  54°  38'  57";  the  angle 
Ajis.   \      opposite,  51°  2'  35";  and  the  angle  opposite 

I      the  qnadrantal  side  72°  26'  21". 
8.  In  a  quadrnTital  triangle,  given  the  qnadrantal  side, 
31 


;62  SPHERICAL    TRIGONOMETRY. 


90°,  and  the  two  adjacent  angles,  one  69°  13'  16",  the 
other  72°  12'  4'',  to  find  the  other  parts. 

C  One  of  the  remaining  sides  is  70°  8'  39".  the 
Ans.   <      other  is  73°  17'  29",  and  the  angle  opposite 

I     the  qiiadrantal  side  is  96°  13'  23". 

4.  In  a  quadrantal  triangle,  given  the  quadrantal  side, 
90°,  one  adjacent  side,  86°  14'  40",  and  the  angle  oppo- 
site to  that  side,  37°  12'  20",  to  find  the  other  parts. 

rThe  remaining  side  is  4°  43'  2";  the  angle  op- 
Ans.   <      posite,  2°  51'  23" ;  and  the  angle  opposite 
I     the  quadrantal  side,  142°  42'  3". 

5.  In  a  quadrantal  triangle,  given  the  quadrantal  side, 
90°,  and  the  other  two  sides,  one  118°  32'  16",  the  other 
67°  48'  40",  to  find  the  other  parts  —  the  three  angles. 

j-The  angles  are  64°  32'  21",  121°  3'  40",  and 
Ans,   <      77°  11'  6" ;  the  greater  angle  opposite  the 
(     greater  side,  of  course. 

6.  In  a  quadrantal  triangle,  given  the  quadranta^  side, 
90°,  the  angle  opposite,  104°  41'  17",  and  one  adjacent 
Bide,  73°  21'  6",  to  find  the  other  parts. 

J        (  Remaining  side,  49°  42'  16"  ;  remaining 
^^*  I      angles,  47°  32'  38",  and  67°  56'  13". 


SOLUTION  OF  OBLIQUE-ANGLED  SPHERICAL  TRIANGLES. 

All  cases  of  oblique-angled  spherical  trigonometry 
may  be  solved  by  right-angled  Trigonometry,  except 
two;  because  every  oblique-angled  spherical  triangle  is 
composed  of  the  sum,  or  the  difference,  of  two  right- 
angled  spherical  triangles. 

WJien  a  side  and  tivo  of  the  angles,  ov  an  angle  and  twit 
of  the  sides  are  given,  to  find  the  other  parts,  conform  to 
the  following  directions : 

Let  a  perpendicular  be  drawn  from  an  extremity  ol  a 
given  si  le,  and  opposite  a  given  angle  or  its  supplement; 
this  will  form  two  right-angled  spherical  triangles :  and 


SECTION  IV.  9C8 

one  of  them  will  have  its  hypotenuse  and  one  of  its  ad- 
jacent angles  given,  from  which  all  its  other  parts  can 
be  computed ;  and  some  of  these  parts  will  become  as 
known  parts  to  the  other  triangle,  from  which  all  its 
parts  can  be  computed. 

To  facilitate  these  computations,  we  here  give  a  sum- 
mary of  the  practical  truths  demonstrated  in  the  fore- 
going propositions. 

1.  The  sines  of  the  sides  of  spherical  triangles  are  propor- 
tional to  the  sines  of  their  opposite  angles. 

2.  The  sines  of  the  segments  of  the  base,  made  by  a  per- 
pendicular from  the  opposite  angle,  are  proportional  to  the 
cotangents  of  their  adjacent  angles. 

3.  The  cosines  of  the  segments  of  the  base  are  proportional 
to  the  cosines  of  the  adjacent  sides  of  the  triangle. 

4.  The  tayigents  of  the  segments  of  the  base  are  reciprocally 
proportional  to  the  cotangents  of  the  segments  of  the  vertical 
angle. 

5.  The  cosines  of  the  angles  at  the  base  are  proportional 
to  the  sines  of  the  corresponding  segments  of  the  vertical 
angle. 

6.  The  cosines  of  the  segments  of  the  vertical  angle  ant 
proportional  to  the  cotangents  of  the  adjoining  sides  of  the 
triangle. 

The  two  cases  in  which  right-angled  spherical  triangles 
are  not  used,  are, 

1st.  When  the  three  sides  are  given  to  find  the  angles ; 
and, 

2d.  When  the  three  angles  are  given  to  find  the  sides. 

The  first  of  these  cases  is  the  most  important  of  all, 
and  for  that  reason  great  attention  has  been  given  to  it, 
and  two  series  of  equations,  (T'and  U,  Prop.  7,  Sec.  Ill), 
have  been  deduced  to  facilitate  its  solution. 

As  heretofore,  let  ABC  represent  any  triangle  whose 
angles  are  d3noted  by  A,  B,  and  C,  and  sides  by  a,  b. 


364  SPHERICAL    TRIGONOMETRY. 

and  c ;  the  side  a  being  opposite  L  ^>  ^^^  side  b  oppo- 
site L  ^>  6tc. 

EXAMPLES. 

1.  In  the  triangle  ABa,a  =  70°  4'  18'' ;  6  =  63°  21'  27" ; 
and  c,  59°  16'  23*^ ;  required  the  angle  A, 

The  formula  for  this  is  the  first  equation  in  group  T, 
Prop.  7,  Sec.  Ill,   which  is 


A  _  /E'  sin.iS'sin.f/S' — aV  J 
2        \         sin. 6  sin.c         / 


"We  write  the  second  member  of  this  equation  thus 


v/" 


(^\  (.^)  (sin.S)  sin.f.S'— a) 


showing  four  distinct  factors  under  the  radical. 

The  logarithm  corresponding  to  - — -.  is  that  of  sm.d 

B 

subtracted  from  10;   and  of  - —  is  that  of  sin.c  sub- 

tracted  from  10,  which  we  call  sin.eomplement. 

BC==a=    70°     4'  18" 

AB=^c=    59°  16'  28"  sin.  com.       .065697 

J[C=  6  =    63°  21'^  27"  sin.  com.       .048749 


2)192°  42'     8" 

S  =     96°  21'     4"  sin. 

9.997326 

,Sf  -  a  =     26°  16'  46"  sin. 

9.646158 

2)19.757930 

U=    40°  49'  10"  COS. 

9.878965 

2 

A  =     81°  38'  20" 

WTien  we  apply  the  equation  to  find  the  angle  A,  we 
/rrite  a  first,  at  the  top  of  the  column ;  when  we  apply 
the  equation  to  find  the  angle  B,  we  write  b  at  the  top 
of  the  column.     Thus, 


SECTION   IV  865 

To  find  the  angle  B. 


sin. a  sin.c 


=  \/Y-^)  (S-)  (Bm.S)  sin.iS^b) 
V        sin.«/  Vsin.c/  ^  ^         ^ 


b  ==  03°  21'  27" 


c  =  59°  16'  23" 

sin.com. 

. 

.065697 

a=  70°     4'   18" 

sin.com. 

. 

.026875 

2)192°  42'     8" 

S=:  96°  21'     4" 

sin.  . 

, 

9.997326 

^Sf— i>=  32°  59'  37" 

sin.  . 

. 

9.736034 

2) 

19.825872 

iB=  35°    4'  49" 

cos.  . 

9.912936 

2 

B  =  70°     9'  38" 

By  the  other  equation  in  formulae  {T,  Prop.  7,  Sec. 
Ill),  we  can  lind  tlie  angle  0;  but,  for  the  sake  of  variety, 
we  will  find  the  angle  (7  by  the  application  of  the  third 
equation  in  formulae  (C/,  Prop.  7,  Sec.  III). 


^_     /R's\Ti.(S—b)  sm.(S—a) 

\  Kin  7i  fiin  ^ 


=v(i£z)( 

c=  59°  16'  23" 
0.=  70°     4'  18" 
b  =  63°  21'  27" 

-^)sin.  (^_t)8in.(^— a; 
sin.ft/           V            >         V 

sin.com.         .026817 
sin.com.         .048479 

2)192°  42'     8" 

^=96°  21'    4" 

^—a  =  26°  16'  46" 

^—6  =  32°  59'  37" 

sin. 
sin. 

.     9.646158 
.     9.736034 

2  )  19.457488 

}C=32°  23'  17" 
2 

sin 

.     9.778744 

C7=64°  46'  34" 
85* 


366  SPHERICAL    TRIGONOMETRY. 

To  sLow  the  harmony  and  practical  utility  of  these  two 
Bets  of  equations,  we  will  find  the  angle  A^  from  the 
equation 

^'"-^^  =\/  (si)  (s-fe)  "''•('^-*)  ^-•('^-'')- 

a  =  70°     4'    18" 

h  =  63°  21'    27"    sin.com.         .048749 


c=  59°  16'  23" 

sin.com. 

.065697 

2)192°  42'  8" 

.S'=  96°  21'  4" 

•>S'— 6=  32°  59'  37" 

sin 

9.736034 

;Sf~-c=  37°  4'  41" 

sin. 

9.780247 

2  " 

)  19.630727 

U  =  40°  49'  10" 
2 

sin. 

9.815363 

A  =  81°  38'  20" 

2.  In  a  spherical  triangle  ABC^  given  the  angle  J.,  38' 
19'  18";  the  angle  B,  48°  0'  10'';  and  the  angle  (7, 
121°  8'  6";  to  find  the  sides  a,  6,  c. 

By  passing  to  the  triangle  polar  to  this,  we  have, 
(Prop.  6,  Sec.  I,  Spherical  Geometry), 

A=  38°  19'  18"  supplement  141°  40'  42" 
j5=:  48°  0'  10"  supplement  131°  59'  50" 
C=   121°     8'     6"  supplement    58°  51'  54" 

We  now  find  the  angles  to  the   spherical  triangle, 
the  sides  of  which  are  these  supplements. 
Thus,      .       141°  40'  42" 


131°  59' 
58°  51' 

50" 
54" 

sin.com. 
sin.com. 

.128909 
.067551 

)  332°  32' 

26" 

166°  16' 
24°  35' 

13" 
31" 

sin.  • 
sin. 

9.375375 

9.619253 

2)19.191088 

66^  47' 

37  r 

cos. 

9  595544 

SECTION  IV.  867 

60°  47'  37  J" 

2r_ 

angle  =  121°  35'  15" 

supp.  =    58°  24'  45"  =  a  of  the  original  triangle. 
IiJ  the  same  manner  we  find  h  =  60°  14'  25";  c  =  89°  1    14". 

It  is  perhaps  better  to  avoid  this  indirect  process  of 
computing  the  sides  of  a  spherical  triangle  when  the 
angles  are  given,  by  the  application  of  the  equations  in 
group  V  or  IF,  Prop.  8,  Sec.  III.  We  will  illustrate 
their  use  by  applying  the  second  equation  in  group  (TF), 
for  computing  the  side  h.     This  equation  is 


fiin  \h—  ( 

-cos.^S'  cos.{S  —  B)\\ 

siu.A  sin.C          / 

A  =    38°  19'  18" 

B  =    48°    0'  10" 

6^=121°    8'    6" 

2  )  207°  27'  34" 
^=103°43'47"--cos.>S'=  -f  Bin.l3°  43'  47"=  9.375376 
B  =    48°    0'  10"     cos.(S-^B)  =  55°  43'  37"=  9.750612 
( S—B)  =    55°  43'  37"  2  )  19.125988 

square  rooi;  =  9.562994 
sin.^  =    38°  19'  18"  =  9.792445 
sin.  C  ==  121°     8'     6"  =  9.932443 
2)19.724888 
square  root  =  9.862444  =  9.862444 
diff.  — 1.700550 
Add  10,  for  radius  of  the  table,     10 
Tabular  sin. ii  =    30°     7'  14"  ^TgiTOOsHo 

2 

h  =  60°  14'  28",  nearly. 

PRACTICAL    PROBLEMS. 

1.  In  any  triangle,  ABC,  whose  sides  are  a,  5,  c,  given 
6  =  118°  2'  14'',  (?  =  120°  18'  33",  and  the  included  angle 
A  =  27°  22'  34",  to  find  the  other  parts. 


368  SPHERICAL    TRIGONOMETRY. 

Ans. 


'■{' 


Ans. 


=  23°  57'  13'',  angle  B  -  91°  26'  44,  and  C  - 

102°  5'  52". 

2.  Given,  ^  =  81°  38'  17",  i?  =  70°  9'  38",  and  Q^ 
64°  46'  32",  to  find  tlie  sides  a,  h,  c. 
^^     r  a  ==  70°  4'  13",  h  =  63°  21'  24",  and  c  =  59°  16' 

8.  Given,  the  three  sides,  a  =  93°  27'  34",  h  --«  100°  t 

26",  and  c  =  96°  14'  50",  to  find  the  angles  A,  B,  and  (7. 

^^     r  J.  =  94°  39'  4",  ^  =  100°  32'  19",  and  (7=  96° 

*  I     58'  35". 

4.  Given,  two  sides,  b  =  84°  16',  e  =  81°  12',  and  the 
angle  (7=  80°  28',  to  find  the  other  parts. 

^The  result  is  amhiguous,  for  we  may  consider 
the  angle  B  as  acute  or  obtuse.  If  the  angle 
B  is  acute,  then  JL  =  97°  13'  45'',- J5  =  83°  11' 
24",  and  a  =  96°  13'  33".  If  ^  is  obtuse,  then 
A  =  21°  16'  43",  B  =  96°  48'  36",  and  a  = 
21°  19'  29". 

5.  Given,  one  side,  (?=64°  26',  and  the  angles  adjacent, 
A  =  49°,  and  B  =  52°,  to  find  the  other  parts. 

A       (h  =  45°  56'  46",  a  =  43°  29'  49",  and  (7=  98<» 
'I      28' 4". 

6.  Given,  the  three  sides,  a  =  90°,  b=  90°,  (j  =  90°,  to 
find  the  angles  A,  B,  and  0. 

Ans.  A  =  90°,  B  =  90°,  and  (7=  dC\ 

7.  Given,  the  two  sides,  a  =  77°  25'  11",  c  =  128°  13' 
47",  and  the  angle  (7  =  131°  11'  12",  to  find  the  other 
parts. 

^^     f  5  =  84«>  29' 20",  A  =  69°  13'  59' and  B  =  72°  28' 

*  I      42". 

8.  Given,  the  three  sides,  a  =  68°  34'  13",  h  =  59° 
21'  18",  and  c  =  112°  16'  32",  to  find  the  angles  A,  B, 
and  a 

.       (A  =  45°  26'  38",  B  =  41°  11'  30' ,  (7  =  134°  53' 


SECTION    IV.  869 

9.  Given,  a  =  89^  2V  37'',  6=  97°  18'  39",  o-=  86°  53' 
46",  to  find  A,  B,  and  C. 

^^^  r  ^  =  88°  57'  20",  B  =  97°  21'  26",  C  =  86°  47' 

10.  Giv^en,  a  =  31°  26'  41",  c  =  43°  22'  13",  and  the 
an^le  ^  =  12°  16',  to  find  the  other  parts. 

j-Amhiguous;  b  =  73°  7'  34",  or  12°  17'  40"; 
AicsJ      angle  5=157°  3'  44",  or  4°  58'  30";  C=  16° 
I     14'  27",  or  163°  45'  33". 

11.  In  a  triangle,  ABC,  we  have  the  angle  J. =56°  18' 
40",  5=  39°  10'  38";  AI),  one  of  the  segments  of  the 
base,  is  32°  54'  16".  The  point  2>  fiills  upon  the  base 
AB,  and  the  angle  C  is  obtuse..  Required  the  sides  of 
the  tiiangle  and  the  angle  C. 

/Ambiguous;  (7=135^25',  or 

135^  57'  ;    c=122^  29',  or 

'   )      123°  19' ;   a=  89°  40',  or 

(        90°  20'  ;    h=  49°  23'  41". 

12.  Given,  A  =  80°  10'  10",  B  =  58°  48'  36",  (7  =  91<» 
62'  42",  to  find  a,  5,  and  c. 

Atu.  a«n79°  38'  22",  5  =  58°  39'  16",  c  =  86°  12'  50". 


B70 


SPHERICAL    TKIGO^^OMETRY, 


SECTION   V, 


APPLICATIONS  OF  SPHERICAL  TRIGONOMETRY  TO 
ASTRONOMY  AND   GEOGRAPHY. 

SPHERICAL  TRIGONOMETRY  APPLIED  TO  ASTRONOMY. 


Sphekical  Trigonometry  becomes  a  science  of  incalcu- 
lable importance  in  its  connection  with  geography,  navi- 
gation, and  astronomy;  for  neither  of  these  subjects  can 
be  understood  without  it ;  and  to  stimulate  the  student 
to  a  study  of  the  science,  we  here  attempt  to  give  him  a 
glimpse  at  some  of  its  points  of  application. 

Let  the  lines  in  the 
annexed  figure  represent 
circles  in  the  heavens 
above  and  around  us. 

Let  Z  be  the  zenith,  or 
the  point  just  overhead, 
Hch  the  horizon,  FZR 
ihe  meridian  in  the  hea- 
vens, and  P  the  pole  of 
the  celestial  equator;  Ph 
is  the  latitude  of  the 
observer,  and  PZ  is  the 

co.latitude.  Qcq  is  a  portion  of  the  equator,  and  tne 
dotted,  curved  line,  mS' S^  parallel  to  the  equator,  is  the 
parallel  of  the  sun's  declination  at  some  particular  time ; 
and  in  this  figure  the  sun's  declination  is  supposed  to  be 
north.     By  the  revolution  of  the  earth  on  its  axis,  the 


SECTIOK    V.  371 

Bun  is  apparently  brought  from  ihe  horizon,  at  S,  to  the 
meridian,  at  m ;  and  from  thence  it  is  carried  down  on 
the  same  curv^e,  on  the  other  side  of  the  meridian ;  and 
this  apparent  motion  of  the  sun  (or  of  any  other  celestial 
body,)  makes  angles  at  the  pole  P,  which  are  in  direct 
proportion  to  their  times  of  description. 

The  apparent  straight  line,  Zc,  is  what  is  denominated, 
in  astronomy,  the  prime  vertical;  that  is,  the  east  and  west 
line  through  the  zenith,  passing  through  the  east  and  west 
points  in  the  horizon. 

"When  the  latitude  of  the  place  is  north,  and  the  decli- 
nation is  also  north,  as  is  represented  in  this  figure,  the 
sun  rises  and  sets  on  the  horizon  to  the  north  of  the  east 
and  west  points,  and  the  distance  is  measured  by  the  arc, 
cSy  on  the  horizon. 

This  arc  can  be  found  by  means  of  the  right-angled 
spherical  triangle  cqS,  right-angled  at  q.  Sq  is  the  sun's 
declination,  and  the  angle  Scq  is  equal  to  the  co.latitude 
of  the  place ;  for  the  angle  Pch  is  the  latitude,  and  the 
angle  Scq  is  its  complement. 

The  side  cq,  a  portion  of  the  equator,  measures  tho 
angle  cPq,  the  time  of  the  sun's  rising  or  setting  beforo 
or  after  six  o'clock,  apparent  time.  Thus  we  perceive  that 
this  little  triangle,  cSq,  is  a  very  important  one. 

When  the .  sun  is  exactly  east  or  west,  it  can  be  deter 

mined  by  the  triangle  ZPS' ',   the  side  PZ  is  known, 

being  the  co.latitude ;  the  angle  PZS'  is  a  right  angle, 

•  and  the  side  PS'  is  the  sun's  polar  distance.    Here,  then, 

'are  the  hj^otenuse  and  side  of  a  right-angled  spherica. 

triangle  given,  from  which  the  other  parts  can  be  com- 

tputed.  The  angle  ZPS'  is  the  time  from  noon,  and  thp 
side  ZS'  is  the  sun's  zenith  distance  at  that  time. 

The  following  problems  are  given,  to  illustrate  ttie 
important  applications  that  can  be  made  of  the  righfr 
angled  triangle  cqS. 


372  SPHERICAL    TRIGONOMETR^i. 

PRACTICAL    PROBLEMS. 

1.  At  what  time  will  the  sun  rise  and  set  in  Lat.  48* 
NT.,  w^hen  its  declination  is  21°  E".? 

In  this  problem,  we  must  make  5'jS'=21°,  P/i=48°=the  angle 
Vch.  Then  the  angle  Scq  =  42°.  It  is  required  to  find  the  arc 
cq,  and  convert  it  into  time  at  the  rate  of  four  minutes  to  a  degree. 
This  will  give  the  apparent  time  after  six  o'clock  that  the  gun  sets, 
and  the  apparent  time  before  six  o'clock  that  the  sun  rises,  (no 
allowance  being  made  for  refraction). 
Making  cq  the  middle  part,  we  have 

R  m\.cq  =  tan. 21°  tan.48° 
tan.21°  =  9.584177 
tan.48°  =  10.045563 


cj=  25°  14'  5"  =  25.2846°         9.629740,  rej. siting  10, 
4 


1*  40"*  56* 
Adding  to  6* 


Sun  sets  p.  M.,      7*  40"*  56',  apparent  time, 

From  6^^ 

Taking  1'^  40"'  56' 


Sun  rises  A.  M.,    4*  19"*    4*,  apparent  time. 
From  this  we  derive  the  following  rule  for  finding  the  apparent 
time  of  sunrise  and  sunset,  assuming  that  the  declination  under- 
goes no  change  in  the  interval  between  these  instants,  which  we 
may  do  without  much  error. 

RULE. 

To  the  logarithmic  tangent  of  the  suns  declination,  add  tht 
logarithmic  tangent  of  the  latitude  of  the  observer  ;  and,  after 
rejecting  ten  from  the  result,  find  from  the  tables  the  arc  of 
which  this  is  the  logarithmic  sine,  and  convert  it  into  time  ai 
the  rate  of  4  minutes  to  a  degree. 

This  time,  added  to  6  o'clock,  will  give  the  time  of  sunset, 
^nd^  subtracted  from  6  o'clock,  will  give  the  time  of  sunrise, 


SECTION    V.  373 

when  the  latitude  and  declination  are  both  north  or  both 
touth ,  but  when  one  is  north,  and  the  other  south,  the  addi- 
tion gives  the  time  of  sunrise,  and  the  subtraction  the  time  of 
sunset, 

2.  At  what  time  will  the  sun  set  when  its  declination 
is  23°  12'  K,  and  the  latitude  of  the  place  is  42°  40'  K? 

Ans.  7^  SS"*  4*,  apparent  time. 

3.  ^Yliat  will  he  the  time  of  sunset  for  places  whose 
latitude  is  42°  40'  N.,  when  the  sun's  declination  is  15° 
21'  south  ?  Ans.  5''  1"*  23*,  apparent  time. 

4.  What  will  be  the  time  of  sunrise  and  sunset  foi 
places  whose  latitude  is  52°  30'  JS".,  when  the  sun's  decli- 
nation is  18°  42'  south  ? 

.         f  Rises  7''  44"*  42',  1  ,  . . 

^^*-   I  Sets    4.  15.  18^;  I  apparent  time. 

5.  What  will  be  the  time  of  sunset  and  of  sunrise  at 
St.  Petersburgh,  in  lat.  59°  56',  north,  when  the  sun's 
declination  is  23°  24',  north?  What  will  be  its  ampli- 
tude at  these  instants  ?  Also,  at  what  hours  will  it  be 
due  east  and  west,  and  what  will  be  its  altitude  at  such 
times  ? 

Sun  sets  at  9*  13'"  30'  p.m.  \  apparent 


Ans. 


'  P.M.  1 
•  A.M.  / 


Sun  rises  at  2"  46"*  30'  a.m.  J       time 
Sun  rises  N.  of  east  1  59°  9f>'  18" 
Sun  sets  N.  of  west   / 
Sun  is  east  at  6*  58"*  2'  a.m. 
Sun  is  west  at  5*  1"*  58*  p.m. 
1^  Alt.  when  east  and  west  is  27°  18'  57". 


UN  THE  AlPLICATrON  OF  OBLIQUE-ANGLED   SPHERICAL 
TRIANGLES. 

One  of  the  most  important  problems  in  navigation 
abd  astronomy,  is  the  determination  of  the  formula  foi 
32 


874 


SPIlEmCAL  TRIGONOMETRY. 


time.  This  problem  will 
be  understood  by  the  tri- 
angle PZS.  When  the 
sun  is  on  the  meridian,  it 
is  then  apparent  noon. 
When  not  on  the  meri- 
dian, we  can  determine 
tl  e  interval  from  noon, 
by  means  of  the  triangle 
FZS\  for  we  can  know 
all  its  sides;  and  the 
angle  at  P,  changed  into 

time  at  the  rate  of  15°  to  one  hour,  will  give  the  time 
from  apparent  noon,  when  any  particular  altitude,  as 
TS^  may  have  been  observed.  P/S'is  known,  by  the  sun's 
declination  at  about  the  time;  and  FZ  is  known,  if  the 
observer  knows  his  latitude. 

Having  these  three  sides,  we  can  always  find  the  sought 
angle  at  the  pole,  by  the  equations  already  given  in 
formulae  (2^,  or  £/,  Prop.  7,  Sec.  Ill);  but  these  formulae 
require  the  use  of  the  co. latitude  and  the  eo.altitude,  and 
the  practical  navigator  is  very  averse  to  taking  the  trou- 
ble of  finding  the  complements  of  arcs,  when  he  is  quite 
certain  that  formulae  can  be  made,  comprising  but  the 
arcs  themselves. 

The  practical  man,  also,  very  properly  demands  the 
most  concise  practical  results.  No  matter  how  much 
labor  is  spent  in  theorizing,  provided  we  arrive  at  prac- 
tical brevity ;  and  for  the  especial  accommodation  of 
.seamen,  the  following  formula  for  finding  time  has  been 
deduced. 

From  the  symmetrical  formulae  {S')  Prop.  T,  Sec.  HI, 
rro  have, 

p       cos.^>S'—  cos.P^  COS. PS 
cos  /  -  -        -^;^pz^,^jys 

Kow,  in  place  of  cos. ZS.  we  take  sin.AS'jT,  which  is,  in 


SECTION    V.  o76 

fact,  the  same  thing ;  and  in  pla«^G  of  cob.PZ,  we  take 
sin.  1  at.,  which  is  also  the  same. 

In  short,  let  A  =  the  altitude  of  the  sun,  L  =  the  la- 
Litude  of  the  observer,  and  D  =  the  sun's  polar  distance. 

rri^  n         SUl.A sin.X  .COS.i> 

Then,      cos.P  = -.,r—, — =- — 

cos.L  Bin.D 

But,    28in.4P  =  1  —  cos.P.      (See  Eq.  32,  Prop.  2, 
Sec.  I,  Plane  Trig.) 

Therefore, 

rt  .    , ,  75      -       sin.J. — sin.X  cos.i) 
28m.»  JP  =  1 .^—. — y- 

cos.i/  sin.i> 

(cos.X  sm.D  +  sin.i  cos.i))  —  sin.^ 

cos.X  sin.i) 
__  sin.(i^  +  B)  —  sin.^ 
coa.Lain.D 
(Considering  (L  +  i>)  as  a  single  arc,  and   (appljmg 
Equation  16,  Sec.  I,  Plane  Trig.),  we  have,  after  dividing 
by  2, 

C08.( -2 )  sin.( ). 

sin.'JP=  5^—: — f{ 

^  cos,L  sin.i> 

^  ,         L  +  D-A^L  +  B+A        . 

But,         2 2 ' 

and  if  we  assume  S= ^ , 

,    ni  •HID      cos.aS' 8in.(AS' — A) 

we  shall  have,  sin.' JP  = =-A — jr — '- 

'  cos.L  sin.D 

or,  Bin.iP  =   s/'lE^^W^. 
'        ^  ^        COS.L  sm.i> 

This  is  the  final  result,  when  the  radius  is  unity ;  when 
^.he  radius  is  i?  times  greater,  then  the  sin.iP  will  be  B 
limes  greater ;  and,  therefore,  the  value  of  this  sine,  cor- 
responding to  our  tables,  is, 

8in.iP  =  v/G^)  (^ji)  co^.SBm.(S-A). 
*  ^    Ncos.i^  \sin.i>^ 


576  SPHERICAL    TRIGONOMETRY. 

PRACTICAL  PROBLEMS. 

1.  In  lat.  39°  6'  20"  I^ortli,  when  the  sun's  declination 
was  12°  3'  10"  :tTorth,  the  true  altitude*  of  the  sun's  cen- 
ter was  observed  to  be  30°  10'  40",  rising.  What  waa 
tlie  api)arent  time  ? 


Alt.  30°  10'  30" 
Lat.  39°  6'  20" 
RD.   77°  56'  50" 

2  )  147°  13'  40" 

S  =  73°  36'  50" 

cos.com.  .110146 
siu.com.  .009680 

COS.    9.450416 

A.)   =  43°  26'  20" 

sin.    9.837299 

30°  22'  5" 

9 

2  )  19.407541 
sin.    9.703770 

R  =  60°  44'  10" 

This  angle,  converted  into  time  at  the  rate  of  15^  to 
one  hour,  or  4  minutes  to  1°,  gives  4*  2"*  56'  from  appa- 
rent noon;  and  as  the  sun  was  rising,  it  was  before 
noon  or 

T  57'"4'A.M. 

If  to  this  the  equation  of  time  were  applied,  we  should 
have  the  mean  time ;  and  if  such  time  were  compared 
with  that  of  a  clock  or  watch,  we  could  determine  its 
error.  A  good  observer,  with  a  good  instrument,  can, 
in  this  manner,  determine  the  local  time  within  4  or  6 
seconds. 

2.  In  lat.  40°  21'  North,  the  true  altitude  of  the  sun,  m 
tho  forenoon,  was  found  to  be  36°  12^,  when  the  decliua- 

*  The  instrument  used,  the  manner  of  taking  the  altitude,  its  cor- 
rection for  refraction,  semi-diameter,  and  other  practical  or  circum- 
stantial details,  do  not  belong  to  a  work  i>f  this  kind,  but  tf  a  work  ob 
Practical  Astronomy  or  Navigation. 


SECTION  V.  377 

tion  of  the  sun  was  3°  20'  South.     What  was  the  appa- 
rent time  ?  Ans.  9"  42'"  40*  a.  m. 

3.  In  latitude  21°  2'  South,  when  the  sun's  declination 
was  18°  32'  North,  the  true  altitude,  in  the  afternoon, 
was  found  to  be  40°  8'.  What  was  the  apparent  time 
of  day  ?  Ans.  2'^  3"*  57"  p.  m. 

SPHERICAL  TRIGONOMETRY  APPLIED  TO  GEOGRAPHY. 

If  we  wish  to  find  the  shortest  distance  between  two 
places  over  the  surface  of  the  earth,  when  the  dis- 
tance is  considerable,  we  must  employ  Spherical  Trigo- 
nometry. 

Suppose  the  least  distance  between  Rome  and  New 
Orleans  is  required;  we  would  first  find  the  distance  in 
degrees  and  parts  of  a  degree,  and  then  multiply  that 
distance  by  the  number  of  miles  in  one  degree. 

In  the  solution  of  this  problem,  it  is  supposed  that  we 
have  the  latitude  and  longitude  of  both  places.  Then 
the  distances,  in  degrees,  from  the  north  pole  of  the 
earth  to  Rome  and  to  New  Orleans  are  the  two  sides  of 
a  spherical  triangle,  the  difterence  of  longitude  of  the 
two  places  is  the  angle  at  the  pole  included  between 
these  sides,  and  the  problem  is,  to  determine  the  third 
side  of  a  spherical  triangle,  when  we  have  two  sides  and 
the  included  angle  given. 

Let  P  be  the  north  pole,  B  the  position  of  Rome,  and 
N  that  of  New  Orleans. 

Lat.  Long. 

New  Orleans,  29°  57'  30"  N.  90°  W. 

Rome,  41°  53'  54"  N.  12°  28'  40"  E. 

Whence,  FE  =  48°  6'    6", 

PiV^  =  60°  2'  30". 

Angle  NFR  =  102°  28'  40". 

32* 


378 


SPHERICAL  TRIGONOxAIETItY. 


We  now  employ  Na- 
pier's 1st  and  2cl  Analo- 
gies, and  find  the  dis- 
tance, in  degrees,  to  be 
78^  48'  15".  This  re- 
duced to  miles,  at  the 
rate  of  69.16   miles  to      I  •  ) 

the  d  ^gree,  will  make      ';  / 

I  he      distance       5450.1       \  / 

miles.  \  / 

The   angle   at   iV  is           ^,  / 

47°  48'  13"  and  at  11,  59°               '^-^.  ^^ 

34'  47''.  ^^- '" 

The  third  side  of  a  spherical  triangle  can  be  found  by 
a  single  formula,  as  we  shall  see  by  inspecting  formulae 
{S')  Prop.  7,  Sec.  III. 

Let  C  be  the  included  angle,  and  c  the  unknown  side 
opposite;  then, 

^        COS. 6?  —  COS. a  cos. 5 

COS.  (7  =  , T—, 

sin. a  sm.o 

Adding  1  to  each  member,  and  reducing,  observing  at 

the  same  time  that  1  -f  cos.  (7=  2cos.'^|(7,  we  have, 

sin.(X  sin. 5  —  cos.a  cos.5 -f  cos.c 


2cos.'^i(7  = 


sin. a  sin.6 


Whence,  2cos.*J(7  sin.a  sin.6  =  cos.c  —  cos.(a-f-5); 
or,  cos.(?  =  COS. (a  +  6)  +  2cos.'^  J(7  sin. a  sin. 5. 

The  second  member  of  this  equation  is  the  algebraic 
sum  of  two  decimal  fractions,  and  expresses  the  value  of 
the  natural  cosine  of  the  side  sought. 

This  case  of  Spherical  Trigonometry,  namely,  that  in 
which  two  sides  and  the  included  angle  are  given,  to 
find  the  third  side,  is  very  extensively  used  in  practical 
astronomy,  in  finding  the  angular  distance  of  the  moon 
from  the  sun,  stars,  and  planets.  For  this  pui'pose,  the 
right  ascension  and  declination  of  each  body  must  be 


SECTION    V. 


879 


found  for  the  same  moment  of  absolute  time.  Their 
difference  in  right  ascen- 
sion gives  the  included 
angle,  P,  at  the  celestial 
pole.  The  declination 
subtracted  from  9.0°,  if  it 
be  north,  and  added  to 
90°,  if  it  be  south,  will 
give  the  sides,  PZ  and 

Pas'. 

In  the  following  exam- 
ples, we  give  the  right 
ascension  and  declination 
of  the  bodies,  and  from 

these  the  student  is  required  to  compute  the  distance 
between  them. 

The  right  ascensions  are  given  in  time.     Their  differ- 
ence must  be  changed  to  degrees  for  the  included  angle. 

MEAN     TIME     GEEENWICH. 

June  24,  1860. 


moon's 

JUPITER'S 

R.A. 

Dec. 

E.  A. 

Dec. 

Distanea. 

h.   m.    8. 

o    /       tr 

h.  m.  8. 

o      /      // 

O        0        If 

A.t  neon,  10  51  36.5 

3  35  24  N. 

8  4  27.6 

20  51  36.8  N. 

44    8  IS 

«  3  h.,    10  58     1 

2  47  43 

8  4  34.2 

20  51  17.8 

45  53  47 

«  6  h.,    11     4  24.6 

1  59  56.2 

8  4  40.8 

20  50  58.7 

47  39  18 

'  8h.,    11  10  47.6 

1  12     6.1 

8  4  47.4 

20  50  39.6 

49  24  4S 

Octohei 

-  6,  1860. 

^R.A. 

Dee. 

0  R.A. 

Dec. 

Distance. 

h.  m.    s. 

o     /      /» 

h.  m.     8. 

o     1      n 

O          /          M 

^t  noon,  5  41  20.8 

26     8     ON. 

12  49  29.3 

5  18  42.6  S. 

107  37    3 

«  3  h.,    6  48  30.1 

26     3  20 

12  49  56.7 

5  21  85.4 

106    8  19 

"  6  h.,    5  55  4« 

25  57  19.4 

12  50  24.1 

5  24  28.2 

104  39  19 

«  9  h.,    6     2  50.5 

25  49  58.1 

12  50  51.4 

5  27  20.9 

103  10     0 

'•12h.,   6  10     1.3 

25  41  15.8 

12  51  19.0 

5  30  13.6 

101  40  Sf 

380  SPHERICAL  TllIGONOMETRY. 


SECTION  VI 


REGULAR  POLYEDRONS 

A  Reg^ar  Polyedron  is  a  polyedron  having  all  its  faces  equa! 
and  regular  polygons,  and  all  its  polyedral  angles  equal. 

The  sum  of  all  the  plane  angles  bounding  any  polyedral  angle  w 
less  than  four  right  angles ;  and  as  the  angle  of  the  equilateral  tri- 
angle is  I  of  a  right  angle,  we  have  §  X  8<4,  |  x  4<^4,  and  |  x  5<^4 ; 
but  I  x  6=4,  I  X  7>4,  and  so  on.  Hence,  it  follows  that  three, 
and  only  three,  polyedral  angles  may  be  formed,  having  the  equi- 
lateral triangle  for  faces;  namely,  a  triedral  angle  and  polyedral 
angles  of  four  and  of  five  faces. 

There  are,  therefore,  three  distinct  regular  polyedrons  bounded 
by  the  equilateral  triangle. 

1.  The  Tetraedron,  having  four  faces  and  four  solid  angles. 

2.  The  Octaedron,  having  eight  faces  and  six  solid  angles. 

3.  The  Icosaedron,  having  twenty  faces  and  twenty  solid  angles. 
With  right  plane  angles  we  can  form  only  a  triedral  angle ;  hence, 

with  equal  squares  we  may  bound  a  solid  having  six  faces  and  eight 
equal  triedral  angles.     This  solid  is  called  the  Hexaedron. 

The  angle  of  the  regular  pentagon  being  f  of  a  right  angle,  we 
have  fx3<;^4;  but  4x4^4;  hence,  with  plane  angles  eqaal  t: 
those  of  the  regular  pentagon,  we  can  form  only  a  triedral  angle. 
The  solid  bounded  by  twelve  regular  pentagons,  and  having  twenty 
solid  angles,  is  called  the  DodecaedroiL 

There  are,  then,  but  five  regular  polyedrons,  viz. :  The  tetraedron^ 
the  octaedron,  and  the  icosaedron^  each  of  whi(jh  has  the  equilateral 
triangle  for  faces ;  the  hexaedron,  whose  faces  are  equal  squares, 
and  the  dodecaedron,  whose  faces  are  equal  regular  pentagons. 

It  is  ob\riou8  that  a  sphere  may  be  circumscribed  about,  or  in- 
scribed within,  any  of  these  regular  solids,  and  converselv :  and 


SECTION  VI 


381 


that  these  spheres  will  have  a  common  center,  which  may  also  be 
taken  as  the  center  of  the  polycdron. 

Anyreirular  polyedron  maybe  regarded  as  made  up  of  a  number 
of  regular  pyramids,  whose  bases  are  severally  the  faces  of  the 
polyedron,  and  whose  common  vertex  is  its  center.  Each  of  these 
pyramids  will  have,  for  its  altitude,  the  radius  of  the  inscribed 
sphere;  and  since  the  volume  of  the  pyramid  is  measured  by  one 
third  of  the  product  of  its  base  and  altitude,  it  follows  (hat  the 
Tolume  of  any  regular  polyedron  is  measured  by  its  surface  multi- 
plied  by  one  third  of  the  radius  of  the  inscribed  sphere. 


PROBLEM. 

Given,  the  name  of  a  recjular  'polyedron,  and  the  side  of  the  hound- 
xng  polygon,  to  find  the  inclination  of  its  faces;  the  radii  of  the  in- 
scribed  and  circumscribed  splicres  ;  the  area  of  its  surface  ;  and  its 
volum  . 

Let  ABha  the  intersection  of  two  adjacent  fiices  of  the  polye- 
dron, and  C  and  D  the  centers  of  these  faces,  0  being  the  center 
of  the  polyedron.  Draw  the  radii, 
00  and  0Z>,  of  the  inscribed,  and 
the  radii  OA  and  0^,of  the  circum- 
scribed sphere ;  also  from  C  and  i) 
let  fall  the  perpendiculars  OE  and 
DE,  on  the  edge  AB,  and  draw  OE; 
then  will  the  angle  DEO  measure 
the  inclination  of  the  faces  of  the 
polyedron,  and  the  angle  DEO  is 
one  half  of  this  inclination. 

Let  1  denote  the  inclination  of  the 
faces,  m  the  number  of  faces  which 
meet  to  form  a  polyedral  angle,  n  the 
nuiibbr  of  sides  in  each  face,  and 
suppose  the  edge  of  the  polyedron  to 
be  unity. 

The  surface  of  the  sphere  of  which  0  is  the  center,  and  radius 
unity,  will  form,  by  its  int(  rsections  with  the  planes,  AOE,  AOD^ 
DOE,  the  right-angled  spherical  triangle  dae,  right-angled  at  e 
in  the  right-angled  triangle  DEO,  the  angle  DOE  is  equal  to 


382  SPHERICAL  TRIGONOMETRY. 

90°  — Z)^0  =  90°  — i/, 
and  is  measured  by  the  arc  de.     The  angle  daCj  of  the  spherical 

kriangle,  is  equal  to ,  and  the  angle  ade  =  ^  ^ 


2m'  °  2n 

Now,  by  Napier's  Rules  we  have 

QO^.dae  =  sin.acZe  cos.c?e. 

or,  cos.rfe  =  _ ;  ( 1 ) 

sin.ade 

and,  cos.CT^  =  cot.f/ae  cot.a^c  (2) 

Substituting  in  eq.  ( 1 ),  for  the  angles  dae  and  ade,  their  values, 

we  find 

cos.360^ 


^m  fox 

-  sin.d60° 


Equation  (3)  gives  the  value  of  the  sine  of  one  half  of  the  incli- 
nation of  the  planes  ;  .and  by  means  of  this  equation  we  may  readily 
find  the  radii  of  the  inscribed  and  circumscribed  spheres. 

In  the  triangle  BED,  we  have 

nE=  BE  cot.BDE  =  icot.  5^, 

2n 
amce  AB  =  1,  and  BE  =  iAB. 
In  the  triangle  DOE,  we  have 

on  =  BE  tan.  U  =  icot.  ^  tan.Ji  (4) 

2n 

From  the  triangle  A  OD,  we  find 

cos.BOA   :   1    ::    OD   :    OA 

whence  OA  =  ?.:?__ 

cos.Z^O^ 

But  the  angle  DOA  is  measured  by  the  arc  ad-,  hence,  substi- 
tuting in  this  last  equation  the  values  of  cos.DOA  and  OD,  takes 
from  eqs.  (2)  and  (4),  we  have 

(?^  =  »tan.i7cot.  ?^  X      ^  ^ 


2n         cot.360°        cot.360' 


2m  2n 

^tan.J/tan.??^°,  (5) 

2m 


by  writing  tan.  for  — ,  and  reducing, 
cot. 


SECTION  IV.  383 

Equation  (4)  gives  the  value  of  OD,  the  radius  of  the  inscribed 
sphere,  and  equation  (5)  gives  that  of  OA^  the  radius  of  the  cir- 
cumscribed sphere.  The  area  of  one  of  the  faces  of  the  polyedron 
is  equal  to  one  half  of  the  apothegm  multiplied  by  the  perimeter. 

360° 

The  apothegm,  as  found  above,  is  equal  to  J  cot. ;  hence,  we 

2n 

onrvo 

ba'e  In  X  \  cot ,  for  the  area  of  one  of  the  faces;  and  multi- 

2/1 

plying  this  by  the  number  of  faces  of -the  polyedron,  we  shall  have 

the  expression  for  its  entire  area.     The  expression  for  the  surface 

multiplied  by  one  third  of  the  radius  of  the  inscribed  sphere,  gives 

the  measure  of  the  volume  of  the  polyedron. 

In  what  precedes,  we  have  supposed  the  edge  of  the  polyedron 
to  be  unity.  Having  found  the  radii  of  the  inscribed  and  circum- 
scribed spheres,  the  surfaces,  and  the  volumes  of  such  polyedrons, 
to  determine  the  radii,  surfaces,  and  volumes  of  regular  polyedrons 
having  any  edge  whatever,  we  have  merely  to  remember  that  the 
homologous  dimensions  of  similar  bodies  are  proportional ;  their 
surfaces  are  as  the  squares  of  these  dimensions ;  and  their  volumes 
as  the  cubes  of  the  same. 

Formula  (3)  gives,  for  the  inclination  of  the  adjacent  faces  of 


The  Tetraedron,         70° 

31^  44^^ 

"    Ilexaedron,         90° 

00^  00^^ 

"    Octaedron,         109° 

28^  18^^ 

"    Dodecaedron,    116° 

33^  54^^ 

"    Icosaedron,       138° 

IV  2y' 

The  subjoined  table  gives 

the  surfaces  and  volumes  of  the  legalav 

oolyedrons,  when  the  edge  is  unity. 

Surfaces. 

Volnmes. 

Tetraedron, 

1.7320508 

0.1178513 

Ilexaedron, 

6.0000000 

1.0000000 

Octaedron,  . 

3.4641016 

0.4714045 

Dodecaedron. 

20.6457288 

7.6631189 

Icosaedron, 

8.6602540 

2.1816%0 

CONTENTS. 

PART    I. 
PLANE    TRIGONOMETEY. 

SECTION    I.  Tagk 

Elementnry  Principles 244 

Dcfiniliuns 24'} 

Propositions 248 

Equations  for  tlie  8inos  of  the  Angles 260 

Natural  ^ines,  Cosines,  etc 265 

Trigonometrical  Lines  for  Arcs  exceeding  9(P 270 

S  ECTION    II. 

Plane  Trigonometry,  Practically  Applied 272 

Logarithms 278 

GENERAL   APPLICATIONS    'tt-ITII   TIIK  rSK    OF    LOGARITHMS. 

L  Eiglit-Anded  Trig«mometry 2S8 

II.  Oblique- Angled  Triironometry 291 

Practical  Problems  295 

SECTION    I  II. 

Application  of  Trigonometry  to  Measuring  Heights  and  Distimces 298 

Practical  Problems 305 


PART    II. 

SPHERICAL   GEOMETRY  AND   TRIGONOMETRY. 

SECTION    I. 

Spherical  Geometry 310 

SECTION    II. 

Eight-Angled  Spherical  Trigonometry 830 

Napier's  Circular  Parts 335 

SECTION    III 

Oblique-Anglod  Spherical  Trigonometry 837 

Napier's  Analogies o48 

SECTION  I  y. 

Spherical  Trigonometry  Applied— Solution  of  Kight-Angled  Spherical  Triangles 353 

Practical  Problems  ^<5 

SHlution  of  Quadrantal  Triangle.^ 358 

Practical  Problems 361 

Solution  of  Oblique-Angled  Spherical  Triangles 362 

Practical  Problems 367 

SECTION    V. 

Spherical  Trigonom'^try  applied  to  Astronomy 870 

Application  of  Obliqne-Anirled  Spherical  Triangles 373 

Spherical  Trigonometry  applied  to  Geography 37T 

Table  of  Mean  Time  at  Greenwich 879 

SECTION    YI. 

Regular  Polyedrons ■< 3S0 


LOGARITHMIC  TABLES; 

ALSO   ▲    TABLE    OV 

NATURAL    AND     LOGARITHMIC 

SINES,  COSINES,  AND  TANGENTS 

TO   EVERY    MINUTE    OF   THE    QUADRANT. 


1 

LOGARITHMS    OF    NUMBERS 

noM 

1    TO    10000. 

N. 
1 

Log. 

N. 

Log. 

i    N- 

Log. 

N. 

Ix,g. 

0  000000 

26 

1  414973     !    61 

1   707570 

76 

1  880814  i 

2 

0  301030 

27 

1  431364     1    52 

1  716003 

77 

1  886491    ! 

3 

0  477121 

28 

1  447158 

63 

1  724276 

78 

1  892095 

4 

0  6020d0 

29 

1    462398 

54 

I  732394 

79 

1  897627 

6 

0  G98970 

30 

1  477121 

55 

1  740363 

80 

1  903090 

6 

0  778151 

31 

1  491362 

56 

1  748188 

81 

1  908485 

7 

0  845098 

32 

1   505150 

57 

1   755875 

82 

1  913«14 

8 

0  903090 

33 

1  518514 

58 

1  ^63428 

83 

1  919078 

9 

0  954243 

34 

1   531479 

59 

1    /70852 

84 

1  924279 

10 

1  000000 

35 

I  544068 

60 

1  778151 

85 

1  929419 

11 

1  041393 

36 

1  556303 

61 

1   785330 

86 

1  934498 

12 

1  079181 

37 

I   568202 

62 

1   792392 

87 

1  939")  19 

13 

1   113943 

38 

1   579/84 

(>3 

1   799341 

88 

1  944483 

14 

1   146128 

39 

1   591065 

64 

1  806180 

89 

1  949390 

15 

1   176091 

40 

1   602060 

(>5 

1   812913 

90 

1  954243 

16 

1   204120 

41 

1   C12784 

66 

I  819544 

91 

1   959041 

17 

1  230449 

42 

1  623249 

67 

1  826075 

92 

I   9«>3Th8   1 

18 

1   255273 

43 

1   633468 

(>8 

1  832509 

93 

1  968483 

19 

1   278754 

44 

1  643453 

69 

1  838849 

94 

1   973128 

20 

1  301030 

45 

1   653213 

70 

1  845098 

95 

1  977724 

21 

1  322219 

46 

1  662758 

71 

1   851258 

96 

1   9S2271 

22 

1  342423 

47 

1  672098 

72 

1   857333 

97 

1   9S(;7.2   1 

23 

1  361728 

48 

1  681241 

73 

1  863323 

i;8 

1   lt9122t)   j 

24 

1  360211 

49 

1  690196 

1    74 

1   8v-9-^32 

y;j 

1   Ui>5i,35    1 

25 

1  397940 

50 

i 

1  698970 

175 

1  8-50ol 

100 

2  000000 

N 

B.  In  the  following  table,  in  the  last  nine  columns  of  each  page,  where 

the  i 

irst  or  leading  figures  change  from  9's  to  O's,  points  or  dots  are  now 

intrc 

)duced  instead  of  the  O's  through  the  rest  of  the  line,  to  catch  the  eye,   j| 

and 

to  indicate  that  from  thence  the  corresponding  natural  number    in    j| 

the  ( 

irst  column  stands  in  the  next  lower  line,  and  :is  annexed  first  two   1 

figUJ 

es  of  the  Logarithms  iu  tlie  secoud  columo. 

I 

i 

LOGARITHMS  OF  NUMBERS.      3 

N. 

0 

1 

2 

3 

4 

5 

6  i 

1 

'  1-" 

9 

3891 

100 

oooono 

0134 

0868 

1301 

1734 

2166 

2598 

3029  3461 

101 

4321 

4750 

5181 

5609 

6038 

6466 

<i894 

7321 

7748 

8174 

102 

8ti00 

9026 

9451 

9876 

.300 

.724 

1147 

1570 

1993 

2415 

103 

012837 

3259 

3680 

4100 

4621 

4940 

63()0 

5779 

6197 

6616 

lOi 

7033 

7451 

7868^ 

8284 

8^00 

9116 

9632 

9947 

.361 

.776 

105 

021189 

1603 

2016 

2428 

2841 

3252 

3664 

4075 

4486 

4896 

10:) 

630') 

5715 

6125 

6533 

6942 

7350 

7767 

8164 

8571  •  8978 

107 

9384 

9789 

.195 

.600 

1004 

1408 

1812 

2216 

2619  3021 

108 

033 124 

3826 

4227 

4628 

5029 

5430 

6830 

6i30^ 

6629  7028 

109 

7426 

7825 

8223 

8620 

9017 

9414 

9811 

.207 

.602 

.998 

110 

041393 

1787 

2182 

C576 

2969 

3362 

3756 

4148 

4540 

4932 

111 

5323 

5714 

6105 

6495 

6885 

7275 

7664 

8053 

8442 

8830 

11-2 

9218 

9606 

9993 

.380 

.  76() 

1153- 

1638. 

1924 

2309 

um 

113 

0530-8 

34()3 

3846 

4230 

4<J13 

4996 

6378 

5760 

6142 

6524 

114 

6905 

7286 

7666 

8046 

8426 

8805 

9186 

9563 

9942 

.320 

115 

030398 

1075 

1452 

1829 

2206 

2582 

2958 

3333 

3709 

4083 

lit) 

4458 

4832 

6206 

5580 

5953 

6326 

6699 

7071 

7443 

7815 

117 

8186 

8557 

8928 

9298 

9o(i8 

..38 

.407 

.776 

1145 

1514 

118 

071882 

2250 

2617 

2985 

3352 

3718 

4085 

4451 

4816 

6182 

119 

6647 

5912 

6276 

i5640 

7004 

7368 

7731 

8094 

8467 

8819 

120 

9181 

9543 

9904 

.266 

.626 

.987 

1347 

1707 

2067 

2/,26 

1-21 

082785 

3144 

3503 

3861 

4219 

4576 

4934 

5291 

'  6647 

6004 

1-22 

63()0 

6^16 

7071 

74-26 

7781 

8136 

8490 

8845 

9198 

9562 

123 

9905 

.258 

.611 

.963 

1315 

1()67  • 

2018 

2370 

2721 

3071 

124 

093422 

3772 

4122 

4471 

4820 

5169. 

5518 

6866 

6216 

6562 

125 

6910 

7257 

7604 

7951 

8298 

8644 

8990 

9335 

9681 

1026 

126 

1003/1 

0715 

1059 

1403 

1747 

2091 

2434 

2777 

3119 

3462 

127 

3804 

4146 

4487 

4828 

5169 

5510 

5K51 

6191 

6531 

6871 

128 

7210 

7649 

7888 

8227 

8565 

8903 

9241 

9579 

9916 

.253 

129 

110590 

0926 

1263 

1699 

1934 

2270 

2606 

2940 

3276 

3609 

130 

3943 

4277 

4611 

4944 

6278 

5611 

6943 

6276 

6608 

6940 

131 

7271 

7()03 

7934 

82(.5 

8595 

8926 

9256 

9586 

9915 

0245 

132 

120574 

0903 

1231 

15(iO 

1888 

2216 

2644 

2871 

3198 

3525 

133 

3862 

41,8 

4504 

4830 

5156 

5481 

6806 

6131 

6456 

6781 

134 

7105 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9690 

..12 

135 

130334 

0fi55 

0977 

1298 

1619 

1939 

2260 

25R0 

2900 

3219 

136 

3539 

3858 

-^177 

4496 

4814 

6133 

5451 

5769 

6086 

6403 

137 

6721 

7037 

7354 

7671 

7987 

8303 

,8618 

S934 

9249 

'  9564 

138 

-9S79 

.  194 

.51)8 

.822 

1136 

1450 

1763 

•2076 

2389 

t  27  (»2 

139 

143016 

3327 

3«,30 

3961 

4263 

4574 

4885 

5196 

5507 

5818 

140 

6128 

6438 

6748 

7058 

7367 

7676 

7985 

8294 

8603 

8911 

141 

9219 

9527 

9H35 

.142 

.449 

.756 

10(i3 

1370 

1676 

1982 

142 

152288 

2594 

'-900 

o205 

:510 

3815 

4120 

4424 

4728 

5032 

143 

5336 

5()40 

6943 

(-24(- 

6649 

6852 

7154 

7457 

7759 

8061 

144 

8362 

S664 

89(i6 

9266 

b6(i7 

9868 

.168 

.469 

.769 

1068 

145 

1  1G1368 

1667 

1967 

2266 

2564 

2863 

3161 

3460- 

3758 

4055 

146 

;   4353 

4(.50 

4947 

5244 

5541 

5838 

6134 

6430 

6726 

7022 

147 

7317 

7613 

7908 

8203 

8497 

8792 

90b() 

9380 

9()74 

99<)8 

148 

not;  62  1  0565 

0S48 

1141 

1434 

1726 

2019 

251 1 

2(;03 

2895 

149 

1 

318ti  1  3478 

37(.9 

40(i0 

4351 

4641 

4932 

5222 

5512 

6S(I2   ■ 

18 


4 

LOGAPIITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

150 

176091 

6381 

6670 

6959 

7-248 

7536 

78-25 

8113 

8401 

8689 

151 

89/7 

92t)4 

9552 

9839 

.126 

.413 

.699 

.986 

1272 

1568 

152 

181844 

2129 

2415 

2700 

2985 

3270 

3655 

3c39 

4123 

4407 

153 

4691 

4975 

5-259 

5542 

5825 

6108 

6391 

tK)74 

6956 

7239 

154 

7521 

7803 

8084 

8366 

8647- 
281 

89-28 

9209 

949C 

9771 

..61 

155 

190332 

0612 

0892 

1171 

1451 

1730 

2010 

2289 

2667 

2846 

156 

3125 

3403 

3681 

3959 

4237 

4514 

•4792 

5069 

5346 

56-23 

157 

5899 

6176 

6453 

6729 

7005 

7281 

',656 

7832 

8107 

8382 

158 

8657 

8932 

9206 

9481 

9755 

..29 

.303 

.677 

.860 

1124 

lo9 

201397 

1670 

1943 

2216 

2488 
273 

2761 

3033 

3305 

3677 

3848 

160 

4120 

4391 

4663 

4934 

5204 

5476 

5746 

6016 

6286 

6566 

161 

6826 

7096 

7365 

.634 

7904 

8173 

8441 

8710 

8979 

9247 

162 

9515 

9783 

..51 

.319 

.586 

.853 

1121 

1388 

1654 

1921 

163 

212188 

2454 

2,20 

2986 

3252 

3518 

3783 

4049 

4314 

4579 

164 

4844 

6109 

5373 

5638 

5902 
264 

6166 

6430 

6694 

6957 

7221 

165 

7484 

7747 

8010 

8273 

8636 

8798 

9060 

9323 

9585 

9846 

166 

220108 

0370 

0631 

0892 

1153 

1414 

1675 

1936 

2196 

2456 

167 

2716 

2976 

3236' 

3496 

3755 

4015 

4274 

4533 

4792 

6051  , 

168 

5309 

5568 

5  26 

6084 

6342 

6600 

6858 

7115 

7372 

7630 

169 

7887 

8144 

8400 

8667 

8913 

257 

9170 

9426 

9682 

9938 

.193 

170 

230449 

0704 

0960 

121p 

1470 

1724 

1979 

2234 

Q488 

2742  , 

171 

2996 

32 -.0 

3504 

37^7 

4011 

4264 

4617 

4770 

5023 

6276 

172 

5528 

5781 

()033 

6285 

6537 

6789 

7041 

7292 

7644 

7796 

173 

8046 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

..50 

.300 

174 

240549 

0799 

1048 

1297 

1546 
249 

1795 

2044 

2293 

2641 

2790 

175 

3038 

3283 

3534 

3782 

4030 

4277 

4525 

4772 

5019 

5266 

176 

6513 

575) 

6006 

6252 

6499 

6745 

6991 

7237 

7482 

7728 

177^ 

7973 

8219 

8464 

8709 

8954 

9198. 

9443 

9687 

9932 

.176 

178 

250420 

0664 

0908 

1151 

1895 

1638 

1881 

2125 

2368 

2610 

179 

2853 

3096 

3338 

3580 

3822 
242 

4064 

4306 

4648 

4790 

6031 

180 

5273 

5514 

5755 

6996 

6237 

6477 

6718 

6958 

7198 

7439 

181 

7679 

7918 

8158 

8398 

8637 

8877 

9116 

9366 

9594 

9833 

182 

260071 

0310 

0548 

0787 

10-25 

1263 

1501 

1739 

1976 

2214 

183 

2451 

2688 

2925 

3162 

3399 

3636 

3K73 

4109 

4346 

4582 

1% 

4818 

5054 

5290 

6526 

5761 

235 

8110 

6996 

6232 

6467 

6702 

6937 

185 

7172 

7406 

7641 

7875 

8344 

8578 

8812 

9046 

9279 

186 

9513 

9746 

9980 

.213 

.446 

.679 

.912 

1144 

1377 

1609 

U, 

271842 

2074 

230.) 

2538 

2770 

3001 

3233 

3464 

3696 

3927 

188 

4158 

4389 

4620 

4850 

5081 

6311 

5542 

5772 

6002 

6232 

189 

6462 

6692 

6921 

7151 

7380 
229 

7609 

7838 

8067 

8296 

8526 

190 

8754 

8982 

9211 

9439 

9667 

9896 

.123 

.351 

.578 

.806 

191 

281033 

1261 

1488 

1715 

1942 

2169 

2396 

2622 

2849 

3075 

192 

3301 

t627 

3753 

3979 

4-205 

4431 

4656 

4882 

5107 

5332 

193 

5557 

5782 

6007 

6232 

6456 

6681 

6905 

7130 

7354 

7578 

194 

7802 

8026 

8249 

8473 

8696 
224 

8920 

9143 

9366 

9589 

9812 

195 

20C035 

0257 

0480 

0702 

0925 

1147 

1369 

1591 

1813 

2034 

196 

2-256 

2478 

2699  2920  | 

3141 

3363 

3584 

8804 

4026 

4246 

197 

44()6 

4687 

4907 

5127 

5347 

6567 

6787 

6007 

6226 

6446 

198 

66(55 

6884 

7104 

73-23 

7642 

7761 

7979 

8198 

8416 

8635 

199 

8853 

9071 

9-289 

9507 

9725 

9943 

.161 

.378 

.595 

.813 

OFNUMBERS               5 

N. 

0 

1 

2 

3 

4   1 

5 

6 

7     8  1   9 

200 

3010.10 

1247 

1464 

1681 

1898 

2114 

2331 

2547  2764  29bi, 

201 

31SJO 

3412 

3628 

3844 

4059  1 

4275 

4491 

47(HJ  1  4921  6136 

202 

5351 

5566 

5781 

699() 

6211  1 

6425 

6(J39 

6854  70 .8  7282 

203 

74iJ{) 

7710 

7924 

8137 

8351 

8564 

8778  '  8991 

9204  1  9447   II 

204 

9630 

9843 

..56 

.268 

.481 

212 

2600 

.693 

.906 

1118 

1330 

1542 

205 

311754 

1966 

2177 

2389 

2812 

3023 

3234 

3445 

3656 

20<) 

3867 

4078 

4289 

4499 

4710 

4920 

5130 

5340 

6551 

5760 

1  207 

5970 

6180 

6390 

6599 

6809 

7018 

7227 

7436  7(>4() 

7854 

208 

8063 

8272 

8481 

8689 

8898 

9106 

9314 

9522  9730 > 

9938 

1  209 

320146 

0354 

0562 

0769 

0977 

207 

1184 

1391 

1698 

1805 

2012 

210 

2219" 

2426 

2633 

2839 

3046 

3252 

3458 

36)5 

3871 

4077 

211 

4282 

4488 

4694 

4899 

5105 

5310 

6516 

6721 

5926 

6131 

212 

6336 

6541 

()745 

6950 

7155 

7369 

7563 

7767 

7972 

8176 

213 

83S0 

85H3 

8787 

8991 

9194 

9398 

9:;0l 

9805 

...8 

.211 

214 

330414 

0617 

0819 

1022 

1225 
202 

1427 

1630 

1832 

2034 

2236- 

215 

2438 

2640 

2842 

3044 

3246 

3447 

3649 

3850 

4051 

4253 

216 

4454 

4655 

4856 

5057- 

6257 

6458 

6658 

6859 

6059 

6260 

217 

64t>0 

6660 

6860 

70i)0 

7*2(i0 

7459 

7659 

7858 

8058 

8257 

218 

8456 

8(i56 

8855 

9054 

9253 

9451 

9660 

9849 

.  47 

.246 

219 

340444 

0642 

0841 

1039 

1237 

1436 

1632 

1830 

2028 

2225 

198 

220 

2423 

2620 

2817 

3014 

3212 

3409 

3606 

3802 

3999 

4196 

221 

4392 

45H9 

4785 

4981  '  5178 

5374 

5570 

5766 

69.>2 

(,157 

222 

6353 

6549 

6744 

6939 

7135 

7330 

7526 

7720 

7916 

8110 

223 

8305 

8500 

8694 

8889. 

9083 

9278 

9472 

9666 

9860 

.  54 

224 

350248 

0142 

0636 

0829 

1023 
193 

1216 

1410 

1603 

1796 

1989 

225 

2183 

2375 

2568 

2761 

2954 

3147 

3339 

3532 

3724 

3916 

22(> 

4108 

4301 

4493 

4685 

4876 

5068 

6260 

6452 

6643 

6834 

227 

6026 

6217 

6408 

6599 

6790 

f98l 

7172 

7363 

7554 

7744 

228 

7935 

8125 

H316 

8506 

8696 

8886 

9076 

9266 

9456 

9646 

229 

9835 

..25 

.215 

.404 

.693 
190 

.783 

.972 

1161 

1350- 

1539 

230 

361728 

1917 

2105 

2294 

2482 

2671 

2859 

3048 

3236 

r424 

231 

3612 

3800 

3988 

4176 

43()3 

4551 

4739 

4926 

6113 

6301 

232 

6488 

5675 

5862 

6049 

6236 

(423 

6610 

6796 

<983 

7169 

233 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8846 

9030 

234 

9216 

9401 

9587 

9772 

9958 
185 

.143 

.328 

.513 

.698 

883 

■  235 

371068 

1253 

1437 

1622 

1806 

1991 

2175 

23^0 

2544 

2:28  1 

236 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382  4565  11 

237 

4748 

4932 

5115 

5298 

5481 

5()64 

5846 

6029  6212  6."^94  1 1 

j  2.J8 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852  i 

8034  8216  II 

1  239 

839i: 

8580 

8761 

8943 

9124 

182 

0934 

930() 

9487 

9668 

9b49 

..30  1 

240 

n-jim  1 

0302 

0573 

0754 

1115 

1296 

1476 

1656  1837 

241 

2017 

2197 

2377 

2567 

2737 

2917 

3097 

3277 

3456  3636 

242 

3815 

3995 

4174 

4'J53 

4533 

4712 

4891 

5070 

5249  5428 

243 

5606 

5785 

5964 

6142 

6321 

6499 

6677 

6856 

7034  7212 

244 

7390 

7568 

7746 

7923 

8101 

178 

9875 

8279 

8456 

8634 

88U  8989 

245 

9166 

9M3 

9520 

9(i98 

..51 

.228 

.405 

.582  .759 

246 

390935 

1112 

1288 

1464 

1641 

1817 

1993 

2169 

2345  2521 

247 

2697 

2873 

3048 

3224 

3400 

3575  V3751 

3926 

4101  4277 

248 

4452 

4ii21 

4802 

4977 

5152 

5326  15501 

6676 

5850  6025 

249 

6199 

6374 

6548 

672:2 

6896 

7071  1  7245 

7419 

7592  7766 

j7- 

LOGARITHMS 

1 

N. 

0 

1 

2 

3 

4 

6 

f 

7 

8 

9 

'250 

397940 

8114 

8287 

84()1 

8634 

88i»H 

hOMI 

9lbl 

9328 

9501 

251 

96/4 

9847 

..20 

.192 

.365 

.538 

.711 

.883 

1056 

1228 

252 

401401 

15<3 

1745 

1917 

2089 

22<)1 

2433 

•^606 

2777 

2949 

253 

3121 

3292 

3464 

S635 

3807 

3978 

4149 

4320 

4492 

4663 

254 

4834 

5006 

5176 

5346 

r5i7 

5688 

5858 

(.029 

6199 

6370 

. 

171 

25E 

6540 

6710 

6881  7051   7221 

73:)  1 

7561   7731 

7901   -^070 

256  , 

8-240 

8410 

8579 

S749  8918 

9087 

i>257  9426  ,  9595   3764 

257 

9933 

102 

.271 

.440  ,  .609 

.777 

946   1114 

1283   1451 

258' 

411620 

1788 

1956 

•..124  ^ 

-293 

2461 

1629 

2796 

2964 

3132 

259 

3300 

ti467 

3635 

3803 

S970 

4137 

4305 

4  472 

4639 

4806 

260 

4973 

140 

5307 

r474 

5641 

6808 

i)974 

6141 

6308 

6474 

2(>1 

6641 

6807 

6973 

7139 

.30a 

<472 

7638 

7804 

7970 

8135 

262 

8301 

467 

8633 

8798 

8964 

L129 

9295 

9460 

9()25 

9791 

263 

9956 

121 

.286 

.451 

.616 

.781 

.945 

1110 

1275 

1439 

264 

421604 

1788 

1933 

-.097 

2261 

2426 

2590 

2754 

2918 

3082 

265 

3246 

3410 

3574 

3737 

3901 

4066 

4228 

4392 

4556 

4718 

266 

4882 

5045 

5208 

371 

.63i 

L697 

5860 

6023 

6186 

6349 

267 

6511 

6674 

6836 

6999 

,161 

7324 

7486 

7648 

7811 

7973 

268 

8135 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

269 

9752 

9914 

..75 

.236 

.398 

.559 

.720 

.881 

1042 

1203 

270 

431364 

1525 

1685 

1846 

2007 

2167 

2328 

2488 

2649 

2809 

271 

2969 

3130 

3290 

34.J0 

3610 

3  770 

3!>30 

4090 

4249 

4409 

272 

4569 

4729 

4S88 

5048 

5207 

53o7 

5526 

5«)85 

6844 

6004 

273 

6163 

6322 

6481 

6640 

6800 

6957 

7116 

7i75 

7433 

7592 

274 

7751 

7909 

8067 

8226 

8384 
158 

8542 

8701 

8859 

9017 

9175 

275 

9333 

9491 

9648 

9806 

9964 

.122 

279 

.437 

.694 

.752 

276 

440909 

1066 

1224 

1381 

1538 

1695 

1852 

20!  )9 

2166 

-323 

277 

2480 

2637 

2793 

2950 

3106 

3263 

c419 

3576 

3732 

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2 

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853 

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1000 

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1102 

1163 

1204 

1254 

1305 

1356 

1407 

854 

1458 

1509 

1560 

1610 

1661 

51 

2169 

1712 

1763 

1814 

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1915 

855 

1966 

2017 

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2220 

2271 

2322 

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2474 

2524 

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4195 

4246 

4269 

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4498 

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4751 

4801 

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5054 

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1014 

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1710 

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1958 

875 

2008 

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2107 

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2603 

26,3 

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2801 

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2901 

2960 

877 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

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878 

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3644 

3693 

3643 

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3742 

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3841 

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3939 

879 

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4038 

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4137 

4186 

4236 

4286 

4336 

4384 

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1 

880 

4483 

4532 

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7041 

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1 726 

17.6 

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1920 

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2066 

2114 

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2260 

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2308 

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2105 

2453 

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2550 

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2792 

2841 

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2938 

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3325 

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19 

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2 

3 

4 

5 

6 

7 

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4918 

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5014 

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5255 

5303 

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5399 

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5543 

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6745 

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7224 

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7320 

7368 

7416 

7464 

7512 

7659 

907 

7607 

7655 

7703 

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7799 

7847 

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7942 

7990 

8o;>8 

908 

8086 

8134 

8181 

8229 

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8325 

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909 

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8860 

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910 

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9089 

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9232 

9280 

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1136 

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1231 

1279 

1326 

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915 

1421 

1469 

1516 

1563 

1611 

1658 

1706 

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1801 

1848 

91G 

1895 

1943 

1990 

2038 

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2132 

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2227 

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2937 

2985 

3032 

3079 

3126 

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3221 

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3316 

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3410 

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3552 

3599 

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3693 

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920 

3788 

3835 

3882 

3929 

3977 

4024 

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4260 

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4401 

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4919 

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5061 

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5155 

923 

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5249 

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5390 

543  7 

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5531 

55/8 

6625 

924 

5672 

5719 

5766 

5813 

5860 

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6001 

6048 

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925 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

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6799 

6845 

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6939 

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7033 

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7220 

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9-28 

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929 

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8062 

8109 

8156 

8203 

8249 

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8436 

930 

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8530 

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8623 

8670 

8716 

8763 

8810 

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931 

8950 

8996 

9043 

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9136 

91o3 

9229 

9276 

9323 

9369 

932 

9416 

9463 

9509 

9556 

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9742 

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9835 

933 

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9928 

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1044 

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1229 

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12y6 

1322 

1369 

1415 

1461 

1508 

1554 

1601 

1647 

1693 

937 

1740 

1786 

1832 

1879 

1925 

1971 

2018 

2064 

2110 

2157 

938 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2673 

2619 

939 

2666 

2712 

2768 

2804 

2851 

2897 

2943 

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3036 

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3128 

3174 

3220 

3266 

3313 

3359 

3405 

3461 

3497 

3643 

941 

3590 

3636 

3682 

3728 

3774 

3820 

3866 

3913 

3959 

4005 

942 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

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4420 

4466 

943 

4512 

4558 

4604 

4650 

4(i96 

4742 

4788 

4834 

4880 

4926 

944 

4972 

6018 

6064 

5110 

6156 

46 

6616 

5202 

5248 

5294 

5340 

6386 

946 

6432 

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5524 

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5799 

5845 

946 

6891 

5937 

5983 

6029 

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6121 

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6212 

6268 

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947 

6350 

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7541 

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19 


20 

LOGARITHMS 

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1 

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3 

4 
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5 

6 

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8454 

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9230 

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9366 

9412 

9457 

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9548 

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9639 

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9730 

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9821 

9667 

9912 

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955 

980003 

0049 

0094 

014C  0185 

0231 

0276 

0322 

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0412 

956 

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0503 

0549 

0594  0o40 

0685 

0730 

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0821 

0867 

957 

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0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

958 

1366 

1411 

1456 

1501 

1.547 

1592 

1637 

1683 

1728 

1773 

959 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

960 

2271 

2316 

2362 

2407 

2452 

2497 

2543 

2588 

2633 

2678 

961 

2723 

2769 

2814 

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2904 

2949 

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3040 

3085 

3130 

932 

3175 

3220 

3265 

3310 

3356 

3401 

3446 

3491 

3536 

3581 

963 

3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

984 

4077 

4122 

4167 

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4257 

43v;2 

4347 

4392 

4437 

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965 

4527 

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4752 

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5067 

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5202 

5247 

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5382 

967 

5426 

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5516 

5561 

6608 

6651 

6699 

5741 

5786 

6830 

968 

5875 

6920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

969 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

970 

6772 

6817 

6861 

6906 

6951 

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7040 

7085 

7130 

7176 

971 

7219 

7264 

7309 

7353 

7398 

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7488 

7532 

7577 

7622 

972 

7666 

7711 

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7800 

7845 

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7934 

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8024 

8068 

973 

8113 

8157 

8202 

8247 

8291 

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8381 

8425 

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8514 

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8648 

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8737 

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975 

9005 

9049 

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9316 

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976 

9450 

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9539 

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9850 

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9939 

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0916 

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1004 

1049 

1093 

1137 

1182 

980 

1226 

1270 

1315 

1359 

1403 

1448 

1492 

1536 

1580 

1625 

981 

1669 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

2023 

2067 

982 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

983 

2554 

2598 

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2686 

2730 

2774 

2819 

2863 

2907 

2951 

984 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

985 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3746 

3789 

3833 

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3877 

3921 

3965 

4009 

4053 

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4141 

4185 

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4317 

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4625 

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4757 

4801 

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4933 

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6152 

989 

5196 

5240 

5284 

6328 

5372 

5416 

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5635 

5679 

5723 

5767 

5811 

5854 

6898 

5942 

5986 

6030 

991 

6074 

6117 

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6249 

6293 

6337 

6380 

6424 

6468 

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6512 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

993 

6949 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

994 

7386 

7430 

7474 

7517 

7561 

44 

7998 

7605 

7648 

7692 

7736 

7779 

995 

7823 

7867 

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7954 

8041 

8085 

8129 

8172 

8216 

996 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

8652 

997 

8695 

8739 

8792 

8826 

8869 

8913 

8956 

9000 

9043 

9087 

998 

9131 

9174 

9218 

9261 

9305 

9348 

9392 

9435 

9479 

9522 

999 

9565 

9609 

9652 

9696 

9739 

9783 

9826 

9870 

9913 

9957   i 

TAIJl.K  II.    I.og.  Sines  and  Taiigenis.  ((P)  Natural  Snes.          21 

0 

S.iie.   ID  10" 

Co>.iiH.  |I).  IO"l  Taiig. 

D  10" 

Coiang. 

N.pine 

N.  COS. 

Neg.iuiiiiite' 

lO.oaQooo! 

0.000000 

Infinite. 

00000 

100000 

60 

1 

u.-io.irZti 

000000 

6.4<;3726 

13.536274 

00029 

KJOOiU 

59 

'2 

7()475.) 

o.))ooo 

764756 

235244 

\ 00058 

100. 10  J 

58 

3 

94(H4? 

OOJOi)0 

940847 

059153 

,000.S7 

lOOO.K) 

57 

4 

7.0)5  786 

000)00 

7.0eJ5786 

12.934214 

OJIH) 

100l»>)0 

5() 

6 

162696 

000000 

162()96 

837304 

100145 

lOOOOiJ 

65 

6 

24  IS  77 

9; 999999 

241878 

758122 

100175 

io:);>oo 

54 

7 

30S824 

999999  ! 

308825 

39 1175 

1 00204 

l(KJ0.)t)153 

,8 

366816 

999999 

366817 

633183 

00233 

100.  too;  52 

'  9 

417968 

99i>999 

417970 

582030 

1  002(i2 

lOOO.K)  51 

10 

463725 

99i)998 

4()3727 

536273 

00291 

lOOIKJO  50 

11 

7.5UoIl8 

9.999998 

7.505120 

12.494880 

<)032( 

99999  49 

i  12 

542906 

999997 

542909 

4570<)1 

00349 

99i)99  48 

13 

577668 

999997 

577672 

422328 

■  00378 

J9999  47 

14 

60.')8o3 

9999<)6 

609857 

390143 

. 00407 

9999!) I  4<) 

15 

639816 

999996  ; 

639820 

360180 

00436 

99!)99 

45 

\H 

667845 

999995 

667849 

332151 

004(55 

9!>999 

44 

1  I' 

6941 73 

999995 

694179 

305821  ; 00495 

99999 

43 

IS 

718997 

999994 

719003 

280997  ; 00524 

99999 

42 

19 

742477 

999993 

742484 

257516!  00553 

99998 

41 

20 

764754 

999993 

764761 

235239;  00582 

99998 

40 

21 

7.7S5943 

9.999992 

7.785951 

12.214049  !:oo()n 

99998 

39 

22 

806146 

999991 

806155 

193845 

' 00640 

99998 

38 

23 

825451 

999990 

8254()0 

174540 

00(i69 

99998 

37 

24 

843934 

999989 

843944 

15(i056 

:  00(398 

99998 

36 

26 

861663 

999988 

861674 

138326 

00727 

99997 

36 

26 

878()95 

999988 

878708 

121292 

0075() 

99997 

34 

27 

895085 

999987 

895099 

104901 

00785 

9!)997 

33 

28 

91  OS  79 

9999S6 

910894 

089106 

00814 

99997 

32 

29 

926119 

999985 

926134 

073866 

00844 

99996 

31 

30 

940842 

999983 

940858 

059142 

00873 

99996 

30 

31 

7.95.J082 

2298 
2227 
2161 

9.999982 

0.2 
0.2 

7.955100 

2298 
2227 
2161 
2098 
2039 
1983 
1930 
1880 
1833 
1787 
1744 
1703 
1664 
1627 
1591 

12.044900 

00902 

99996 

29 

32 

9(>8870 

999981 

968889 

031111 

j 00931 

99996 

28 

1  .-Kl 

982233 

999980 

982253 

017747 

00960 

99995 

27 

134 

995198 

999979 

0.2 

995219 

004781 

00989 

99995 

26 

,  3r, 

8.0J7787 

2098 

999977 

0-2 
0-2 
0-2 

8.007809 

11.992191 

01018 

99995 

25 

1  3(i 

0-20021 

2039 

999976 

020045 

979955 

01047 

99995 

24 

1  37 

031919 

1983 

999975 

031945 

968055 

01076 

99994 

23 

38 

043501 

1930 
1880 
1832 
1787 
1744 
1703 
1664 
1626 

999973 

0-2 
0-2 
0-2 
0-2 

043527 

956473'  01105 

99994 

22 

39 

054781 

999972 

054S()9 

945191  |i  01 134 

99994 

21 

40 

0.i5776 

999971 

0(i5806 

9341941  01164 

99993 

20 

41 

8.076500 

9.999969 

8.076531 

11.923469  01193 

99993 

19 

42 

a86965 

9999()8 

0'2 

086997 

913003 

012-22 

99993 

IB 

43 

097183 

999966 

0'2 
02 

o;3 

097217 

902783  i 

01251 

99992 

17 

44 

107167 

999964 

107202 

892797  i 

01280 

99992 

16 

46 

116926 

999963 

1169(S3 

883037 

01 30-^ 

99991 

15 

46 

126471 

1591 
1557 
1524 
1492 
1462 

999961 

0.3 

126510 

873490;  0133*^ 

99991  14 

.  47 

135810 

999959 

03 
0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.3 

135851 

1557 
1524 
1493 
1463 
1434 
1406 
1379 
1353 
1328 
1304 
1281 
1269 
1238 
1217 

864149 

01367 

9999 li  3  i 

48 

144953 

999958 

144996 

855004 

01396 

99990 

12 

;  49 

153907 

99995(> 

153952 

846048! 

014-25 

99990 

11  1 

;  50 

162681 

999954 

162727 

837273 ' 

01454 

99989 

10 

,  51 

8.171280  ;■;;',? 

!™iJ   3  9 
18/980  ,„ 

9.99!)952 

8.171328 

11.828672 

01483 

99989 

9 

152 

999950 

179763 

820237 : 

01513 

99989 

8 

1  63 

999948 

188036 

811964' 

01542 

99988 

7 

54 

196102  ^e;';J  ;  999946 
204U70  l^;^.       99!>944 
211895  ^^^'  999942 
219581  ;*?/  •  999940 

227134   .;:?•;    999938 
23455  7,  .;:;:    999936 

196156 

803844 

01571 

99988 

6 

55 

204126 

795874  01600 

99f>87 

6 

56 

0.3 
0.4 
0.4 
n  A 

211953 

788047 : 

01629 

99987 

4 

57 

219641 

780359 , 

01658 

9998(i 

3 

58 

227195 

772805 

01687 

9998<) 

2 

59 

^l       234621 
^■^       241921 

765379 

01716 

99985 

1 

60 

241855  i'^'*^ 

fOH  If 

9!)9934 

768079" 

01745 

99985 

0 

S  :;•' 

1  Cn-nng. 

Tan?.    N.  ros. 

\.  .«^ine- 

S9  D-grees. 



22 


Log.  Sines  and  Tangents.    (1°)     Natural  Sines.       TaBLK  11. 


08 

1 


4 
5 
6 
7 
8 
9 
ID 
11 
12 
13 
,4 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
!  50 
,51 
I  52 
i53 
64 
55 
56 
57 
68 
59 
60 


.241855 


249033  l]^^ 
256094  }}^' 
2iS3042  ^8 
269881  ^ 
276)14 
. 283243 
289773 
296-207 
30254 
30879 


D  10 


1122 
1105 
1088 
1072 
1056 
,  1041 
7  1027 


.dl49o4  ,f),r, 

321027  ^^^J 
327016  III 
332924  07? 
338753  ^'^ 
344504 
350181 
355783 


361315  Qifj 


366777 
.372171 
377499 
382762 
387962 
393101 


959 
946 
934 
922 


899 
888 
877 
867 
856 

398179  007 
4031991  007 
4081611  ofA 

.422/17:  70, 

427462!  ioi 

432156}  ^°4 

436800  '^^ 

441394  !^ll 

445941  1  '^2 

450440  7?.2 

15^1^35 

4o9301  i  ,727 

463665  70,, 

.467985  '^To 

472263  7A;; 

476498  '"° 

480693  l^. 

484848  ^of^ 

488963  TZa 

493040  ^1^ 

497078  ^^; 

501080  ^^J 

505045  ^"^ 

,508974  III 

']?4l  643 
016/26  j:.„_ 

520551  ^^' 

524343  ^^^ 

528102  l^\ 

S"S  616 
53oo23  ^. . 

539186  l;,t^ 

642819  606 


Cosine. 


.999934 
999932 
999929 
999927 
999925 
999922 
999920 

-999918 
999915 
999913 
999910 

9.9999D7 
999905 

'999902 
999899 
999897 
999894 
999891 
999888 
999885 
999882 

1.999879 
999876 
999873 
999870 
999867 
999864 
999861 
999858 
999854 
999851 

K 999848 
999844 
999841 
999838 
999834 
999831 
999827 
999823 
999820 
999816 

).  999812 
999809 
k'999805 
999801 
999797 
999793 
999790 
999786 
999782 
999778 
99977. 
999769 
999765 
999761 
999757 
999753 
999748 
999744 
999740 
999735 


D.IO^' 

0.4 
0.4 
0.4 
0.4 
0.4 
0.4 


0-4 
0.4 
0.5 
0.5 
0.5 
0.5 
0  5 
05 
0.6 
0.6 
0.6 
0.6 
0.6 
0.6 
0.5 
0.5 
0.6 
0.5 
0.6 
06 
0.6 
0.6 
0-6 
0.6 
0.6 
0.6 
0.6 
0.6 
0.6 
0.6 
0-6 
0.6 
0.6 


0.7 
0.7 
0.7 
0.7 


Tan^r. 

.241921 
249102 
256165 
263115 
269956 
276691 
283323 
289856 
296292 
302634 
308884 
.315046 
321122 
327114 
333025 
333856 
344610 
360289 
355895 
361430 
366895 
.372292 
377622 
382889 
388L192 
393234 
398316 
403338 
408304 
413213 
418068 
.422869 
427618 
432315 
436962 
441560 
446110 
450613 
455070 
459481 
463849 
.468172 
472454 
476693 
480892 
485050 
489170 
493250 
497293 
501298 
505267 
.509200 
513098 
516961 
520790 
624586 
528349 
532080 
535779 
539447 
543084 


D  10' 


1197 
1177 
1158 
1140 
1122 
1105 
1089 
1073 
1057 
1042 
1027 
1013 
999 
985 
972 
959 
946 
934 
922 
911 
899 
888 
879 
867 
857 
847 
837 
828 
818 
809 
800 
791 
783 
774 
7()6 
758 
760 
743 
736 
728 
720 
713 
7U7 
700 
693 
686 
680 
674 
668 
661 
655 
650 
644 
638 
633 
627 
622 
616 
611 
606 


Coiiii)^'. 


Co 


!aiig. 


N.  sine;.  N.  cos 


11 


11  .7.58079 
750898 
743835 
736885 
730044 
723309 
716677 
710144 
703708 
697366 
691116 
684954 
678878 
672886 
666975 
661144 
655390 
649711 
644105 
638570 
633105 

11 .627708 
622378 
617111 
611908 
606766 
601685 
596662 
591696 
586787 
581932 

11.577131 
672382 
567685 
563038 
558440 
553890 
549387 
544930 
540519 
636151 

11.531828 
527546 
523307 
519108 
514950 
510830 
506750 
602707 
498702 
494733 
490800 
486902 
483039 
479210 
476414 
471651 
467920 
464221 
460553 
466916 


P 01742 
01774 
:;  01803 
!  01832 
I  01862 
01891 


99985  60 
)i)984  69 
999S4  5S 
99983  57 
99983  66 
99982  66 


!  I  01949 
ji  01978 
i02U07 
I  02036 

I  02065 
;  02094 
i,  02123 
j;  02152 
ii  02181 
I' 02211 

I I  02240 
'I  02269 
!  02298 
II  02327 
j' 02356 
1102385 
:,  02414 
;j  02443 
I  02472 
I  02501 

02530 

02560 

02589 

'  02618 

H 02647 

i  02676 

!  02705 

02734 

02763 

02792 


::  02850 
1 102879 
1!  02908 
:' 02938 


01920  99982  54 
99^81  63 
99980,  68 
99980  51 
99979  50 
9y979  49 
y9978  48 
99977  47 
9^)977,  46 
99976'  45 
99976  44 
99975  43 
99974!  42 
99974'  41 
99973^  40 
9y972!  39 
99972  38 
99971  37 
99970  36 
y9y69!  35 
99969  34 
99968J  33 
999671  32 
99966;  31 
99966  30 
99965  29 
99964  28 
99963!  27 
99963!  26 
999o2!  25 
99961;  24 


02821  99960!  23 


99959I  22 
99959  21 
99958; 20 
99957'  19 


11 


!  I  02967  99956  I8 
'!02b9b  99955  17 
03025  99954:  16 
03054  99953  15 
( 03083  99952:  14 
i  031 12  99^52  13 
103141  99951;  12 
;  03170  99950!  11 
103199  99949110 
103228  999481  9 
03257  999471 
99946! 
9J9i5 
99944 
99943 
99942 
99941 
99940 
99939 


0328(i 
03316 
jl03<i45 
1 1  03374 
I  03403 
,03432 
!  I  03401 
! 03490 


Tail"-.   i!  N.  COS.  N 


88   DegreeB. 


TABI.K  II.        Log.  Siiit.»  and  Tangents.     (i°)     Natural  Sines. 


23 


.542819 
54G4^>2 
54999.5 
553589 
557054 
6(i0540 
6G3999 
5«)74:n 
67083(> 
574214 
5775(i() 

.5H()89-2 
584193 

•  5874ti9 
590721 
593948 
597152 
600332 
603489 
606623 
609734 

.612823 
615891 
618937 
6219(>2 
624965 
627948 
630911 
633854 
d.3()776 
63^680 

.642563 
645428 
648274 
651102 
653911 
656702 
659475 
662230 
664968 
667(i89 

1.670393 
673080 
675751 
678405 
681043 
683665 
686272 
688863 
691438 
693998 

!.69t)543 
699073 
701589 
704090 
706577 
709049 
711507 
713952 
716383 
718800 


D.  10' 


r/nn    9. 


603 
595 
591 
586 
681 
576 
572 
5(i7 
5()3 
559 
554 
550 
546 
542 
538 
534 
530 
526 
522 
519 
515 
511 
508 
504 
501 
497 
494 
490 
487 
484 
481 
477 
474 
471 
468 
465 
4(i2 
459 
45(> 
453 
451 
448 
445 
442 
440 
437 
434 
432 
429 
427 
424 
422 
419 
417 
414 
412 
410 
407 
405 
403 


Cosine. 

.999735 
999731 
999726 
995)722 
999717 
999713 
999708 
9997t)4 
999699 
9!)9694 
999689 

.999()85 
!W9<)80 
9<)9()75 

"999670 
999665 
9996t)0 
999(555 
999650 
999645 
999640 

.999(135 
999629 
999324 
999{J19 
9f)96l4 
999608 
999()03 
999597 
999592 
999586 

.999581 
999575 
999570 
9995()4 
999558 
999553 
999547 
9})9541 
9J)9535 
999529 

.999524 
999518 
999512 
999506 
999500 
999493 
999487 
99iJ481 
999175" 
939469 

•.9!;9+()3 
9.'i9456 
999 150 
999143 
999437 
999431 
9994-:4 
999418 
9i>9411 
999404 


D.  10' 


0.7 
0.7 
0.7 
0-8 
0-8 
0-8 
0.8 
0.8 
0.8 
0-8 
0.8 
0-8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.9 
0.9 
0.9 
0.9 
0.9 
0-9 
0-9 
0.9 
0.9 
0.9 
0.9 
0.9 
0.9 
0.9 


Sii 


1.0 
1.0 
1-0 
1-0 
1.0 
1.0 
1.0 
1.0 
1.0 
1.0 

1. 

1 

1, 

1, 

1. 

1, 

1 

1, 

I 


Tang. 


.543084 
54(>()91 
5502ti8 
553817 
557336 
560828 
564291 
667727 
571137 
574520 
577877 

,581208 
584514 
587795 
591051 
594283 
597492 
600677 
603839 
606978 
610094 

.613189 
616262 
619313 
622343 
625352 
628340 
631308 
634256 
637184 
640093 

.642982 
645863 
648704 
651537 
654352 
657149 
659928 
662689 
6(i5433 
668160 

.670870 
673563 
67()239 
678900 
681544 
684172 
6  6784 
689381 
691963 
694529 

.697081 
699617 
702139 
704246 
707140 
709«)18 
702083 
714534 
71(>y?2 
719396 


Coiiuit 


D.  10' 


602 
59.) 
591 
587 
582 
577 
573 
668 
664 
559 
555 
551 
547 
543 
639 
635 
53  r 
527 
523 
619 
616 
612 
508 
505 
601 
498 
495 
491 
488 
485 
482 
478 
475 
472 
469 
466 
463 
460 
457 
464 
453 
449 
446 
443 
442 
438 
436 
433 
430 
428 
425 
423 
420 
418 
415 
413 
411 
408 
406 
404 


Colling.  i|N.  Bine.lN  cos, 


11.456916 

453309 
449732 
446183 
442(i64 
439172 
435709 
432273 
4288()3 
425480 
422123 

11.418792 
415486 
412205 
408949 
405717 
402508 
399323 
396161 
393022 
389906 

11.386811 
383738 
380687 
377657 
374648 
3716(J0 
368692 
365744 
362816 
359907 

11.357018 
354147 
351296 
348463 
345648 
342851 
340072 
337311 
334567 
331840 

11.329130 
326437 
323761 
321100 
318456 
315828 
313216 
310()19 
308037 
30.5471 

11.302919 
3fj0383 
297861 
295354 
292860 
290382 
287917 
2854()5 
283028 
280ti04 


;■  03490199939  60 

;  0351;)  99938  59 
,03548 199937  58 
03577 
:  0360() 
i  03635 


9993( 
99935 
99934  55 
03664  99933  54 
(t3(J93 199932  53 
03723  99931 
0375299930 
03781  {99929 
03810  99927 
03839,99926 
(J38ti8 ,99925 


I  03926 '99923 
103955199922 


1O39.S4I9992I 


04013 
04042 
04071 
04100 
03129 
04159 


99919 
99918 
99917 
99916 
99915 
99913 


04188  99912 
•04217199911 
1 04246  J999 10 
04275199909 
0430499907 
04333  !9990e) 
04362)99905 
0439 1|9  9904 
;  0442099902 
04449199901 
0447H|'999(J0 
04507  [99898 
04536 199897 
0456.>9989li 
04594:99894 
04623  y9S9o 
04()53]99892 
04(i82 '99890 
047 1 1  i99889 
04740  [;J9d8h 
047(i9rJ9SH() 
0479hl9.J885 
;04S2/7J9883 
0485()!9JJ8h2 
04H85I99881 
049141998 


24 

,23 

)-7 


'  0494.') 
049  i  2 

! 05001 
05030 
0505!) 
05088 
05117 


9.9878 
99876 
99875 
99873 
99872 
99870 
998(i9 


()5146  9-j8()7 


0517; 


9986(i 


05205  998()4 


05234 


Tant 


N.  cofi.  X.sint' 


998()3 


87  Degrees. 


24 


Log.  Sines  and  Tangeiils.     {^)     Natural  Sines.        TABLE  II. 


0 
1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
J7 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
60 
61 
52 
53 
64 
65 
56 
57 
68 
69 
60 


Sine, 

.718800 
721204 
723595 
725972 
728337 
730088 
733027 
735354 
737667 
739989 
742259 

.744536 
746802 
749055 
751297 
753528 
755747 
757955 
760151 
762337 
764511 

.766675 
768828 
770970 
773101 
775223 
777333 
779434 
781524 
783605 
785675 

.787736 
789787 
791828 
793859 
795881 
797894 
799897 
801892 
803876 
805852 

,807819 
809777 
811726 
813667 
815599 
817522 
819436 
821343 
823240 
825130 

.827011 
828884 
830749 
832607 
834456 
836297 
838130 
839956 
841774 
843585 

Cosine. 


D.  10' 

401 
398 
396 
394 
392 
390 
388 
386 
384 
382 
380 
378 
376 
374 
372 
370 
368 
366 
364 
362 
361 
359 
357 
355 
353 
352 
350 
348 
347 
345 
343 
342 
340 
339 
337 
335 
334 
332 
331 
329 
328 
326 
325 
323 
322 
320 
319 
318 
316 
315 
313 
312 
311 
309 
308 
307 
306 
304 
303 
302 


Cosiuu. 

9.999404 
999.'W8 
999391 
999384 
999378 
999371 
999364 
999357 
999350 
999343 
999336 

9.999329 
999322 
999315 
999308 
999301 
999294 
999286 
999279 
999272 
999266 

9.999257 
999250 
999242 
999235 
999227 
999220 
999212 
999205 
999197 
999189 

9.999181 
999174 
999166 
999158 
999150 
999142 
-999134 
999126 
999118 
999110 

9.999102 
999094 
999086 
999077 
999069 
999061 
999053 
999044 
999036 
999027 
999019 
999010 
999002 
998993 
998984 
998976 
998967 
998958 
998950 
998941 


Sine. 


D.  1 


Tanji. 


S. 7 19396 
721806 
724204 
726588 
728959 
731317 
7336()3 
735996 
738317 
740626 
742922 
.745207 
747479 
749740 
751989 
764227 
766453 
758i)68 
760872 
763065 
765246 
,767417 
769678 
771727 
773866 
775995 
778114 
780222 
782320 
784408 
786486 

8.788554 
790613 
792662 
794701 
796731 
798752 
800763 
802765 
804858 
806742 
,808717 
810683 
812641 
814589 
816529 
818461 
820384 
822298 
824205 
826103 
.827992 
829874 
831748 
833613 
835471 
837321 
839163 
840998 
842825 
844644 


Cot.in2. 


D.  10"      Cotanj?.    |(N.  sine.  N.  cos. 


402 
399 
397 
395 
393 
391 
389 
387 
385 
383 
381 
379 
377 
375 
373 
371 
369 
367 
366 
364 
362 
360 
358 
356 
355 
353 
351 
350 
348 
346 
345 
343 
341 
340 
338 
337 
335 
334 
332 
331 
329 
328 
326 
325 
323 
322 
320 
319 
318 
316 
315 
314 
312 
311 
310 
308 
307 
306 
304 
303 


11.280604 
278194 
275796 
273412 
271041 
2(i8683 
266337 
264004 
261683 
259374 
257078 

11.254793 
252521 
250260 
248011 
245773 
243547 
241332 
239128 
236935 
234754 

11.232583 
230422 
228273 
226134 
224006 
221886 
219778 
217680 
215592 
213514 

11.211446 
209387 
207338 
205299 
203269 
201248 
199237 
197235 
195242 
193258 

11.191283 
189317 
187359 
185411 
183471 
181539 
179616 
177702 
175796 
173897 

11.172008 
170126 
168252 
166387 
164529 
162679 
160837 
159002 
157175 
1563.56 


05234  99863 


05263 
05292 
05321 
05350 
05379 


05408  99854 


05437 
1 105466 
j 05495 
i  05524 
I  05653 
j 105582 
l!  05611 
!  05640 
I' 05669 
1105698 
{!  05727 
I  06766 

I  i 05785 

I I  05814 

I  j 05844 
1106873 
1 105902 

I I  05931 
ij  05960 

{05989 
!  106018 
1106047 
i I  06076 
li  06106 
1106134 
i!  06163 
;|  06192 
1 1  08221 
i  106260 
i^  06279 
jj  06308 
N 06337 
i,  06366 


99861 
99860 
99868 
99857 
99855 


99852 
99851 
99849 
99847 
99846 
99844 
99842 
99841 
99839 
99838 
99836 
99834 
99833 
99831 
99829 
99827 
99826 
99824 
99822 
99821 
99819 
99817 
99815 
99813 
99812 
99810 
99808 
99806 
99804 
99803 
998OI 
99799 
99797 


06395'99795 
06424  997931 
O6453I99792 
{)<i482i99790 
0651199788 
■  06640!99786 
I  0856919978 1 
1 0659819978; 


08627 
06656 
j  06685 
106714 


997SC' 
99778 
99776 
99774 


106743199772 
I  06773  99770 
p  06802  99768 
i' 06831 199766 
;' 06860  99764 
1,06889  99762 
::  069 18  99760 
,i  06947  99758 
'!  06976  99756 


Tang. 


N.  COS.  N'.sine 


60 

59 

58 

57 

56 

55 

54 

53 

52 

51  ' 

50  ; 

49  i 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

16 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


TABLK  II.     .   Log.  Sines  and  Taiigeiils.     (4°)     Naiuiul  Sines. 


25 


18.843585 
84.53^7 
84i-!H3 
848971 
8507.^)1 
85'i5"J5 
854-291 
8.)t.U49 
857801 
.io954() 
8()1-J83 

8.8ii3014 
8(^4738 
8G()455 
%81(i5 
8t;9S(;8 
871.'j<i5 
873255 
874938 
87t)tJl5 
878t2«5 
879949 
881»)07 
883'i58 
884^)03 
88G54-2 
888174 
889801 
8914'->1 
893035 
8941)43 

8.89{>-.'4(i 
89784-2 
899432 
901U17 
902596 
9041(i9 
90573«i 
907297 
908853 
910404 

8.911949 
913488 
915022 
9l»i5;j0 
918073 
919591 
9':  1103 
922(il0 
924112 
925(i0}) 
;  927100 
928587 
9300-i8 
931544 
933015 
934481 
935942 
937398 
938860 
94t/29U 
CoMJnu. 


D.  lU' 

300 
29i) 
298 
2*7 
2)5 
294 
293 
292 
291 
2  0 
288 
287 
286 
285 
284 
283 
282 
281 
279 
279 
277 
276 
275 
274 
273 
272 
271 
270 
269 
268 
2()7 
26() 
265 
264 
263 
262 
2S1 
2(i0 
259 
258 
257 
257 
256 
255 
254 
253 
252 
251 
250 
249 
249 
248 
247 
246 
245 
244 
243 
243 
242 
241 


Co.'sine. 

1.95)8941 
998932 
99S923 
998914 
998.905 
99889(> 
998887 
998878 
998869 
99S860 
99S851 

1.998841 
998832 
998823 
998813 
998S(M 
998795 
998785 
998776 
998766 
99H757 

I  998747 
998738 
998728 
998718 
998708 
998()99 
998(i89 
998679 
})98(;()f> 
9!)8()59 

1.998649 
998(i3*> 
998629 
998619 
998(i09 
998599 
99H589 
998578 
998568 
998558 

1.998548 
998537 
998527 
.998516 

-99850() 
998495 
998485 
998474 
9984(i4 
998453 

•.998442 
998431 
9;<8421 
99S410 
i)98399 

• f 98388 
9i)8377 
998366 
998355 
99>>344 


D.  10" 


1 

1 

1 

I 

1 

1 

1 

1 

1 

1 

1 

I 

1 

1 

1.6 

1.6 

1.6 

1.6 

1.6 

1.6 

1.6 

1.6 

1.6 

1.6 

1.6 

1-6 

1.6 

1-6 


1.6 

1.7 

1.7 

1.7 

1.7 

1.7 

1.7 

1.7 

1.7 

1.7 

1.7 

1.7 

1.7 

1 

I 

1 


7 

7 

7 

•1.8 


1.8 
1.8 


1.8 
1.8 
1.8 
1.8 


Taug. 

.844644 
846455 
84.vJ(il) 
h50>)57 
851846 
853628 
855403 
857171 
858932 
8(J0i;86 
8<>2433 

;  864173 
8t)590t) 
867632 
869351 
871064 
872770 
8744()9 
87(il()2 
877849 
879529 

l.h81202 
882869 
884530 
886185 
887833 
889476 
891112 
892742 
8943(i() 
8!)5984 

;.  897596 
899-203 
900803 
902398 
903987 
905670 
907147 
908719 
910285 
911846 

;. 913401 
914961 
916496 
918034 
919668 
921096 
922619 
924136 
925()49 
927 156 

1.928«i58 
930155 
931647 
933134 
934()16 
93<)()93 
937665 
939032 
940494 
941!j52 

ColJing. 
56  Degreen. 


30-2 
301 
299 
298 
29, 
29  i 
295 
293 
292 
29! 
-290 
289 
288 
287 
285 
284 
283- 
282 
281 
280 
279 
278 
277 
276 
276 
274 
273 
272 
271 
-270 
-269 
268 
267 
266 
'265 
264 
263 
262 
261 
260 
259 
258 
257 
256 
256 
255 
254 
253 
252 
251 
250 
249 
249 
248 
247 
24() 
245 
244 
244 
243 


Cotang.   N.  sine.lN.  rort.l 


11.155366 
153545 
151740 
149943 
148154 
14(i372 
144597 
142829 
141068 
139314 
137567 

11.135827 
134094 
132368 
130649 
l-:-936 
127-230 
125531 
123S38 
122151 
120471 

11.118798 
117131 
115470 
113815 
112167 
110524 
108888 
107258 
105634 
104016 

11.102404 
100797 
099197 
097602 
006013 
094430 
092853 
091281 
089715 
088154 

11,086599 
085049 
083505 
081966 
080432 
0.8904 
077381 
075864 
074361 
072844 

11.071342 
069845 
0<)8353 
0{)6866 
0(i5384 
0»j3907 
062435 
0(i09(i8 
05956t> 
058048 
I'ang. 


(J697(j 
07005 
07034 
07063 


1 99756  60 
99754  59 
99752  58 
; 99750  57 
070<)2;99748  5(i 
07121,99746:55 
07150  99744  54 
07 179 [99 742,  53 
07208  iK)740:  52 
07237, !W738i  51 
07-266'99<3<i  50 
07295  99734  49 
;  07324  99731  i  48 
t  99729  47 


07353 
0*382 


997271  46 
6741199726145 
07440!y"723,  44 
07469:99721  43 
07498199719,  42 
07527  99716  41 
07556199714140 
07585J99712!39 
0761499710  38 
07643  99708  37 
07672  9;)705  36 
07701  99703  35 
077.30  99701  34 
07759  99(i99  33 
07788:99696  32 
07817i99(i94  31 
0^846  99692.  30 
07875  99()89|  29 
07904i^J9o87|  28 
07933 199685;  27 
'■  07962  99(i83:  -26 
07991  |99{i80  -25 
080201)9678,  24 
08049  J99676  23 
08078  99673  '22 
08107^9671  21 
08136|9f>()«;8  -iO 
081(i5jiW666:  19 
08194  99()()4  18 
08-223  99661'  17 
08252199659  16 
08-281 1996571  15 
08310  99654'  14 
08339  9.9652  13 
0H3f, 8  991,49  12 
08397  ^)9(f47l  1 1 
08426  9i)()44  10 
0845.^i99(i42i    9 


08484 
0H513 
08542 
08571 
08()00 
U8()29 
08(i58 
08(i87 
08716 


99()39: 

:>9637 
99(i35 
99()32 
99(>30 

9:,()V7 
99(i25 
9M>22 
99619 


N.  ros.  N'.fine.l 


26 


Log.  Sines  and  Tangents.    (5°)    Natural  Sines.  *■   TABLE  II. 


2 
3 

4 
6 
6 

7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
26 
26 
27 
28 
29 
30 
31 
32 
33 
34 
36 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
4() 
47 
48 
49 
50 
61 
52 
53 
64 
55 
56 
i~i 
68 
59 
60 


Sine. 

,940296 
941738 
943174 
944606 
946034 
947466 
948874 
950287 
951ii9o 
95310a 
964499 
,955894 
957284 
958;)70 
960052 
9ol429 
962801 
934170 
965534 
96()893 
968249 
969600 
970947 
972289 
973628 
974962 
976293 
977619 
978941 
980259 
981573 
982883 
984189 
985491 
986789 
988083 
989374 
990)60 
991943 
993222 
994497 
995768 
997036 
998299 
999560 

'.00J816 
0J20;i9 
003318 
0045()3 
005805 
007044 

I  003278 
009510 
01073  7 
0lia'>/. 
01318--^ 
0144:)0 
015613 
016824 
018031 
019235  j 

Cosine.  | 


D.  10' 


8. 


240 
239 
239 
238 
237 
I  236 
235 
235 
234 
233 
232 
232 
231 
230 
229 
229 
228 
227 
227 
226 
225 
224 
224 
223 
222 
222 
221 
220 
220 
219 
218 
218 
217 
216 
216 
215 
214 
214 
213 
212 
212 
211 
211 
210 
209 
209 
208 
208 
207 
2ij6 
206 
205 
206 
204 
203 
203 
2J2 
202 
201 
201 


Cosine. 


9,998344 
998333 
998322 
998311 
998300 
998289 
998277 
998266 
998265 
998243 
998232 

9.998220 
998209 
998197 
998186 
998174 
998163 
998151 
998139 
998128 
998116 

9.998104 
998092 
998080 
998068 
998056 
998044 
998032 
998020 
998008 
997996 
997984 
997972 
997959 
997947 
997936 
997922 
997910 
997897 
997886 
997872 

9  997860 
997847 
997835 
997822 
997809 
997797 
997784 
997771 
997758 
997745 
997732 
997719 
997706 
997693 
997680 
997667 
997664 
997641 
997628 
997614 


D.  10' 


1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

i.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.2 

2.2 

2.2 

2.2 

2.2 

2.2 


Sine. 


TanS_ 

8.941952 
943404 
944852 
946295 
947734 
949168 
950597 
952021 
953441 
954856 
956267 

8.957674 
959075 
960473 
961866 
9{j3266 
964639 
966019 
967394 
968766 
970133 

8.971496 
972865 
974209 
975560 
976906 
978248 
979586 
980921 
982251 
983577 

8.984899 
986217 
987532 
988842 
990149 
991451 
992750 
994045 
995337 
996()24 
997908 
999188 

9.0i)0465 
001 738 
003007 
004272 
005534 
006792 
008047 
009298 

9.010546 
011790 
013031 
014268 
016502 
016732 
017959 
019183 
020403 
021620 


D.  10' 


Co  tang. 


242 

241 

240 

240 

239 

238 

237 

237 

236 

235 

234 

234 

233 

232 

231 

231 

230 

229 

229 

228 

227 

226 

226 

226 

224 

2iJ4 

223 

222 

222 

221 

220 

220 

219 

218 

218 

217 

216 

216 

215 

215 

214 

213 

213 

212 

211 

211 

210 

210 

209 

208 

208 

207 

207 

206 

206 

205 

204 

204 

203 

203 


Cotanj^.   N.  sine 


11.058048 
066696 
056148 
053705 
0522()6 
050832 
049403 
047979 
046669 
045144 
043733 

11.042326 
040925 
039527 
038134 
036745 
035361 
033981 
032606 
031234 
029867 

11.028504 
027145 
025791 
024440 
02S094 
021752 
020414 
019079 
017749 
016423 

11.015101 
013783 
012468 
011168 
009861 
008649 
007260 
005956 
004663 
003376 
002092 
000812 

10.999535 
998262 
996993 
995728 
99W<)6 
993208 
991053 
990/02 

10.989454 
988210 
686969 
985732 
984498 
983268 
983041 
980817 : 
979697  i 
978380 


|i  08716 

ii  08745 

108774 

ii  08803 

i' 08831 

: 08860 

08889 

,08918 

08947 

!  08976 

09005 

09034 

09063 

09092 


cos 


99619 
99617 
99614 
99612 
99()09 
99607 
99()04 
99602 
99599 
9959(1 
99594 
99691 
99588 
99586 


0912199583 
I  09160199580 
'09179  99578 
109208199576 
:09237!99572 
;  09266199570 
j  0929o!99567 
'  09324i99564 
\  09353199562 
i  09382199559 
;  0941 1199556 
1 09440J99563 
1094(1999.551 
j  09498|99548 
i  09527199646 


i  0955(JJ99642  31 
09685199540 
09614199537 
09()42;99534 
0?>67 199531 
09700199528 
09729  99526 
06<  68  99523 
09787  9!)520 
0981(,9951 
09816  99514 
09874  99511 
09903  9f)508 
09932  9950 
0996199503 
0.>99t)995U0 
10019  99497 
10i>48  99494 
10077  99491 
1010:)994.S8 
10135  99485 
10164  99482 
10192  994,9 
10221994/6 
10-60  99473 
10279  99470 
10308  99467 
10337199464 
10366  9£Kil 
10395  99458 
10424  99,55 
10453  991.52 


Tan-?.        :  N.  cop.  N. sine. 


St  Df^srreeR. 


-j 


TABLE  II. 


Log.  Sinee  and  Taugente.    (6^)    Natural  Sines. 


Tang.     iD.  10"|    Cotang.     iN.  sine. 


0 

1 
Q 
3 
4 
5 
6 
7 
8 
9 
\Q 

i  i: 
1-: 

13 

14 
15 
l(i| 
17 
18 
19 
20 
21 
22 
23 
24 
25 
2d 
27 
28 
29 
30 
31 
32 
33 
34 
36 
3(i 
37 
38 
39 
40 
41 
42 
43 
44 
45 
40 
47 
48 
19 
50 
51 
52 
63 
64 
6.- 
6- 
67 
58 
58 
(>0 


ID.  10' 


.019235 
020435 
021()32 
022825 
0241)1  () 
025203 
02()38t» 
C>275()7 
028744 
029918 
031089 
032257 
^31421 
/J4582 
035741 
03()89G 
038048 
039197 
040342 
041485 
042(>25 

.043702 
044895 
04(i02() 
047154 
048279 
049400 
050519 
051035 
052749 
053859 

. 054900 
050071 
057172 
058271 
059307 
(Ki04(i0 
001651 
0;)2(i39 
0(i3724 
0()480(i 

.005885 
0t>«)902 
008030 
0)9107 
070170 
071242 
0-2300 
0733(;() 
074424 
0754.^0 

I  070533 
0i7583 
078031 
079070 
O.S0719 
08^59 
0.vii97 
083832 
0818()4 
085«94 

CosinH. 


200 
199 
199 
198 
198 
197 
197 
196 
190 
195 
195 
194 
194 
193 
192 
192 
191 
191 
190 
190 
189 
189 
180 
188 
187 
187 
186 
186 
185 
185 
184 
184 
184 
183 
183 
182 
182 
181 
181 
180 
l.'^O 
179 
171- 
179 
178 
178 
177 
177 
176 
176 
175 
175 
175 
174 
174 
173 

i;3 

li2 
172 
172 


Cosine.     D.  10' 


9.997614 
997601 
997588 
997574 
997561 
997547 
997534 
997620 
997507 
997493 
W97480 

9.997466 
997452 
997439 
997425 
-^»7411 
997397 
997383 
997309 
997355 
997341 

9.997327 
997313 
997299 
997286 
997271 
997267 
997242 
997228 
997214 
997199 
997185 
997170 
997166 
997141 
997127 
997112 
997098 
997083 
997008 
997(>53 
.997039 
997024 
997009 
99109941 
996979  j 
996964 ] 
990949  ! 
990934 
990919 
996J»04 

9.990889 
990874 
996858 
990843 
996828 
996812 
<n)()797 
99()782 
99()766 
996751 


2.2 

2.2 

2.2 

2.2 

2.2 

2.2 

2.3 

2.3 

2.3 

2-3 

2-3 

2.3 

2-3 

2.3 

2.3 

2.3 

2-3 

2-3 

2-3 

23 

2.3 

2.4 

2-4 

2.4 

2.4 

2-4 

2.4 

2.4 

2.4 

2.4 

2-4 

2-4 

2.4 

2.4 

2-4 

2.4 

2.4 

2.4 

2.6 

2.5 

2 

2 

I  2 

l2 

;  2 
2 

2-6 
2.6 
2.6 
3.6 
2.5 
2.6 
2.6 
2.5 
2.5 
2.6 
2  6 
2  0 
2  6 
2  6 


Sine. 


9.021620 
022834 
024044 
025251 
02(i455 
027055 
028852 
030046 
031237 
032425 
033009 
034791 
035969 
037144 
038316 
039485 
040051 
041813 
042973 
044130 
046284 

9.04(^434 
047582 
048727 
049869 
051008 
052144 
053277 
054407 
055535 
05<)(>59 
067781 
058900 
000016 
001130 
002240 
01)3348 
004453 
005556 
006666 
067762 

i9.(>68846 
(169038 
071027 
072113 
073197 
074278 
075356 
070432 
077505 
078576 

9.079044 
080710 
081773 
082833 
083891 
084i)47 
t)8(i000 
087050 
088098 
089144 
Cotang. 


202 
202 
201 
201 
200 
199 
199 
198 
198 
197 
197 
196 
190 
196 
196 
194 
194 
193 
193 
192 
192 
191 
191 
190 
190 
189 
189 
188 
188 
187 
187 
186 
186 
186 
186 
185 
184 
184 
183 
183 
182 
182 
181 
181 
181 
180 
IbO 
179 
179 
178 
178 
178 
177 
177 
176 
176 
175 
175 
176 
174 


10.978380 
977100 
975950 
974749 
973545 
973345 
971148 
909954 
908763 
967675 
966391 

10.966209 
904031 
9()2850 
901()84 
900616 
959349 
958187 
957027 
955870 
954716 

10.953560 
952418 
951273 
960131 
948992 
947850 
94^i723 
945593 
944405 
943341 

10.942219 
941100 
939984 
938870 
937760 
936652 
935547 
934444 
933345 
932248 

10.935154 
930062 
928973 
927887 
926803 
925722 
9-2'i644 
923668 
922496 
921424 

10.920356 
919290 
918227 
917107 
9l(;i09 
915053 
914000 
912950 
911J02 
91086() 
Tang."" 


0453 
Olh'J 
0511 
054(1 
05()9 
0597 
062() 
0()65 
0')84 
vAI3 
0742 
0771 
08<H) 
0829 


N.  cog. : 


19452 .  60 

;!)449:59 
<):)44()  58 
99443  67 
)9440  I  5() 
99437  I  65 
9i)434  64 
99431  I  63 
9^-428  u-2 
99424  I  61 
99421160 
9y418 '  49 
1^9416:48 
99412147 
0858{9y4W) ;  46 
<J887I9940()I45 


09 1() 
0945 
0!)73 


99402 1 44 
99399 ! 43 
99396  42 
1002199393  i  41 
1031199390  40 
1000;99a86'39 
1089199383  38 
1118199380137 
1147  99377!  36 
1170  99374135 
1205|993T0  34 
1234;{'9367i33 
1263!J)9364  32 
129ll99300:31 
1320l99i57  130 


1349 
1378 
1407 
143t) 
1465 
1494 
1523 
1552 
1.580 
1609 
1038 
1()67 
169() 
1726 
1754 
1783 
1812 
1840 
18()9 
189K 
1927 
195(. 
19vS5 
2014 
204;i 
2071 
'21()(( 
212!) 
2158 
218; 


99354 ! 29 
99351 : 28 
)9347  i  27 
>9o44  ;  20 
99341  25 
99337  i  24 
99334  23 
99331  I  ii2 
95)327  i 
99324  I 
99320  I 
99317  I 
99314 
99310' 
99307 
99303 
f)9300 


99297 
)9293 
99290 
99286 
9{)283 
(9279 
99276 
)9272 
)92()9 
992  .5 
ii92{i2 
99268 
>f)255 


N  COS.  N.sinf. 


83  D«irree-g. 


28 


Log,  Sines  and  Tangents.    (7°)    Natural  Sines. 


TABLE  II. 


Sine. 
9.085894 

08794  7 
088970 
089990 
09IO;)8 
^•2024 
093037 
094047 
09505t) 
096062 

9.097065 
098056 
099066 
100062 
101056 
102048 
103037 
104025 
105010 
105992 

9.106973 
107951 
108927 
109901 
110873 
111842 
112809 
113774 
114737 
115698 
1 16656 
117613 
118567 
119519 
120469 
121417 
122362 
123306 
124248 
125187 


0 

1 

2 

3 

4 

5 

b 

7 

8 

9 
10 
11 
12 
13 
14 
16 
16 
17 
18 
19 
^20 
21 
22 
23 
24 
26 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41  19.126126 


42  I 

t3l 

44 

45  I 

46 

47 

48 

4P 

51  i 

51  9 

52 

53 

64 

65 

66 

67 

6S 

59 

60 


127060 
127993 
128^26 
129tt64 
130781 
131706 
132630 
133551 
1344/0 

. 135387 
136303 
137216 
138128 
139037 
139944 
140850 
141754 
142655 
143556 

Cosit!^. 


w-w 


171 
171 
170 
170 
170 
169 
169 
168 
168 
168 
167 
167 
166 
166 
166 
166 
166 
164 
164 
164 
163 
163 
163 
162 
162 
162 
161 
161 
160 
160 
160 
159 
169 
169 
168 
168 
168 
157 
167 
157 
166 
166 
166 
156 
166 
164 
164 
154 
153 
153 
153 
152 
162 
152 
152 
151 
151 
151 
160 
150 


Cosiuf. 

9.996761 
996735 
996720 
996704 
996688 
996673 
996657 
996641 
996626 
996610 
996594 

9.996678 
996562 
996546 
996530 
996514 
996498 
996482 
996466 
996449 
996433 
996417 
996400 
996384 
996368 
996361 
996335 
996318 
996302 
996286 
996269 

9.996252 
996236 
996219 
996202 
996186 
996168 
996161 
996134 
996117 
996100 
99()083 
996066 
996049 
996032 
996015 
996998 
995980 
995963 
99594() 
995928 
995911 
995894 
995876. 
995859 
995841 
995823 
995806 
995788 
995771 
995753 


Sine. 


D.   lu' 


2.0 
2.6 
2.6 
2.6 
2.6 
2.6 
2.6 
2.6 
2.6 
2.6 
2.6 
2.7 
2.7 
2.7 
2.7 
2.7 
2.7 
2.7 
2.7 
2.7 
2.7 
2.7 
2.7 
2.7 
2.7 
2.7 
2.7 
2.7 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 


Tang. 

.089144 
090187 
091228 
092266 
093302 
094336 
095367 
096395 
097422 
098446 
099468 

.IOL'487 
101504 
102519 
103532 
104542 
105650 
106656 
107559 
108660 
109559 

.110556 
111661 
112643 
11363a 
114621 
115507 
116491 
117472 
118462 
119429 

. 120404 
121377 
122348 
123317 
124284 
125249 
126211 
127172 
128130 
129087 

.130041 
130994 
131944 
132893 
133839 
134(84 
136(26 
136  567 
137d06 
138542' 
1894/6 
140409 
141340 
142269 
143196 
144121 
145044 
145966 
146885 
14/803 


Lotiing. 


174 
173 
173 
173 
172 
172 
171 
171 
171 
170 
170 
169 
169 
169 
168 
168 
168 
167 
167 
166 
166 
166 
166 
166 
165 
164 
164 
164 
163 
163. 
162 
162 
162 
161 
161 
161 
160 
160 
160 
159 
159 
159 
158 
158 
158 
157 
167 
157 
166 
166 
166 
156 
155 
155 
154 
154 
154 
153 
153 
163 


C(4aiig. 

10.910856 
909813 
908772 
907734 
906698 
905664 
904633 
903605 
902578 
901564 
900532 

10.899613 
898496 
897481 
896468 
896458 
894450 
893444 
892441 
891440 
890441 

10.889444 
888449 
887467 
886467 
885479 
884493 
883509 
882628 
881548 
880571 

10.879596 
878623 
877652 
876683 
876716 
874751 
873789 
872828 
871870 
870913 

10.869959 
86900P 
868056 
867107 
866161 
866216 
864274 
8633;i3 
862396 
861458 

10.860624 
859591 
868660 
857731 
856804 
855879 
854956 
854034 
853116 
852197 


N.  siiii! 

1218/- 

12216 

; 12245 

12274 

12302 

12331 

12360 

I  12389 

: 12418 

i 12447 

! 12476 

I  12504 

12633 


99255 
99251 
99248 
99244 
99240 
99237 
99233 
99230 
9226 
99222 
99219 
99215 
99211 
125(i2 199208 
12.591 199204 
12620^99200 


12649 
12678 
12706 
12735 
12764 
12793 
12822 
12851 
12880 


99197 
99193 
99189 
99186 
99182 
99178 
99176 
99171 
99167 


12908  99163 


12937 
129(]6 
12995 
13024 
13053 
13081 
13110 
13139 
13168 
13197 
13226 
13254 


99160 
99156 
99152 
99148 
99144 
99141 
99137 
99133 
99129 
99125 
99122 
99118 


13283199114 
13312  99110 


13341 


99106 


13370  99102 
13(;99 199098 


1:13427 

I  i  1345b 

i  13i85 

;  13514 

l!  13543 

|il35'/2 

13(00 

13629 

13658 


13687  99059 
13711.99055 


Tang. 


13744 
137-3 

13802 
13831 
13860 
13889 
1391. 
N.  ros. 


99094 
99091 
99087 
99083 
990/9 
990*5 
J90/1 
99067 
99063 


99051 
i9047 
J9043 
99039 
99035 
1903 1 
j9027 


82  Degrees. 


Log.  Sines  and  Tangents.  (8°)  Natural  Sii 


29 


0 

1 

3 
4 
6 
6 

7 
8 
9 

10 

11 

12 

13 

14 

15 

in 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

3-2 

33 

34 

35 

3<i 

37 

38 

39 

40 

41 

42 

43 

44 

46 

4(3 

47 

48 

49 

50 

51 

62  I 

53 

54 

55 

56 

51 

58 

69 

60 


bine. 

43555 
44453 
45349 
l4(i24J 
47136 
48026 
48915 
49802 
5U()86 
51,>69 
52451 
53.i30 
54208 
55i)83 
55957 
56830 
57;O.J 
58569 
59435 
60301 
61164 
62026 
62885 
63743 
[64600 
(J5454 
66307 
67159 
l()8008 
68856 
69702 
70547 
71389 
72230 
73070 
73908 
74744 
755  i^8 
7«)411 
77242 
78072 
78900 
79726 
80551 
181374 
82196 
83016 
83834 
84651 
854l)(i 
8()280 
8/092 
87903 
88712 
895 19 
90326 
91130 
191933 
92734 
93534 
94332 
Cos  in  I'. 


D.  10' 

150 
149 
149 
149 
148 
148 
148 
147 
147 
147 
147 
146 
146 
146 
145 
145 
145 
144 
144 
144 
144 
143 
143 
143 
142 
142 
142 
142 
141 
141 
141 
140 
140 
140 
140 
139 
139 
139 
139 
138 
138 
138 
137 
137 
137 
137 
136 
136 
136 
136 
136 
135 
135 
135 
134 
134 
134 
134 
133 
133 


Cosine.  |D.  10" 

.995753 

9fJ5735 

995717 

995()99 

995681 

995(ili4 

995646 

995(i28 

995610 

995591 

995573 
1.995556 

995o37 

995619 

995601 

995482 

995464 

995446 

995427 

995409 

995390 
.995372! 

995353 ; 

J95334 , 

995316, 

995297 

995278 

995260 

995241  ! 

995222 

995203 

995184! 

995165 

995146, 

995127} 

995108 

995089 

99.5070 

995051 

995032 

995013 
.994993 

994974 

994955 

994935 

994916 

994896 

994877 

994857 

994838 

994818 
.994i98 

994779 

994769 

994739 

994719 

994700 

994(i80 

994660 

99  4640 

994620 
~Sine! 


3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 


Tang. 

.147803 
148718 
149(i32 
150544 
151454 
152363 
1.532(i9 
154174 
156077 
155978 
156877 

.157776 
16867 1 
159565 
160457 
161347 
162236 
163123 
164008 
164892 
165774 

. 166654 
167532 
1684<J9 
169284 
17015/ 
1710-29 
171899 
17276/ 
173634 
174499 

.175362 
176224 
177084 
177942 
178799 
179655 
180508 
181360 
182211 
183059 

. 183907 
184752 
185597 
18li439 
187280 
188120 
188958 
189794 
190629 
191462 

. 192294 
193124 
193953 
194780 
196()06 
196430 
19;253 
198074 
198894 
199713 


Colang. 


D.  10' 


153 
152 
152 
152 
151 
151 
151 
150 
150 
150 
IlO 
149 
149 
149 
148 
148 
148 
148 
147 
147 
147 
146 
146 
146 
145 
145 
145 
145 
144 
144 
144 
144 
143 
143 
143 
142 
142 
142 
142 
141 
141 
141 
141 
140 
140 
140 
14(.» 
139 
139 
139 
139 
138 
138 
138 
138 
137 
137 
1^7 
137 
136 


Cotang.  I  N.  sine.lN.  c««.| 


10.852197 
851282 
850368 
849456 
848546 
847637 
846731 
845826 
844923 
844022 
843123 

10  842225; 
841329 
840435 
839543 
838653 : 
837764  ; 
836877! 
835992 
835108 
834226 

10.833346 

832468 

831591 

830716 

829843 

828971 

828101  ' 

827233 ' 

8263661 

825501  ' 

10.824638  I 
823776! 
822916 
822058! 
821^01 
820345  ' 
819492; 
8186401 
817789 
816941  [ 

10-816093 
8152481 
814403! 
813561 ' 
812720  • 
811880 
811(542' 
81020() 
809371 
80o538 , 

10.807706', 
806876 i  I 
806047; 
805220 ' 
804394 
803570 i 
802747  ' 
801926 
801106 
800287 


3917|99027  60 
394(i;  99023  69 
39/599019  58 
40m!99015  67 
4033;9i>()ll  66 
4061  !»90iH)  55 
4090  99002  54 
4119198998  53 
4148198994  62 
41 77 198990  51 
42059H986  50 
42; ;4. 98!  182  \  49 
4263  9S978  |  48 
-I2;i2'98973  |  47 
43209h9(i9  46 
4349i989()5 
4378'98961 
440/198957 


443()98953 
4464  98948 


4493 
4522 
4561 
4580 
4608 
4637' 


98944 
9h940 
98936 
98931  37 
9h927  36 
98923  '  35 
4666198919;  34 
4695198914  133 
4723i98910!32 


4752 

4781 

4810 

4838 

48(>7 

489() 

4925 

4954 

4982 

5011 

5040 

5069 

5097 

6126 

5155 

5184 

5212 

5241 

5270 

5299 

532 

6356 

5385 

5414 

5442 

5471 

5500 

5529 

5657 

6586 

5615 

5643 


98906131 
)b902  !  30 
9o897 ' 29 
98«93 ' 28 
9b8b9 \  27 
98884 ! 26 
98880 • 25 
L>8876  24 


Tang. 


J8871 
98867 
988()3 
9h858  20 
9hh64  19 
98849!  18 
98845' 17 
98841  I  16 
J8836  i  16 
98832  i  14 
98827  I  13 
98823 '  12 


98818 
98814 
98809 
98805 
98800 
i879«i 
98791 
J8787 
98782 
98778 
98773 
y8769 


N.  COS.  N.fline. 


81  Degrees. 


'30 


Log,  Sines  and  Tangents.  (9°)  Natural  Sines. 


TABm  II. 


Sine.   D.  10' 


7 
8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

2.3 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41  19 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 
'  54 
■  55 
i  56 
l57 
I  58 

59 

60 


.194332 
195129 
19.5925 
196719 
197511 
19830-2 
199091 
1998  79 
20i).)66 
201451 
2U-223-i 

.2J3017 
203797 
204577 
205354 
2C»613l 
206906 
207679 
208452 
209222 
209992 

.210760 
211526 
212291 
2130.55 
213818 
214579 
21.5338 
216097 
216854 
217609 

.218363 
219116 
219868 
220618 
221367 
222115 
2228<)1 
223606 
224349 
225092 

.225833 
226573 
227311 
228048 
228784 
229518 
230252 
230984 
231714 
232444 

.233172 
233899 
234625 
235349 
236073 
236795 
237515 
238235 
238953 
239670 


Co.sine. 


133 
133 
132 
132 
132 
132 
131 
131 
131 
131 
130 
130 
130 
130 
129 
129 
129 
129 
128 
128 
128 
128 
127 
127 
127 
127 
127 
126 
126 
126 
126 
125 
125 
125 
125 
125 
124 
124 
124 
124 
123 
123 
123 
123 
123 
122 
122 
122 
122 
122 
121 
121 
121 
121 
120 
120 
120 
120 
120 
119 


Co.sine.  D.  lU 

.994620 

994600 

994580 

994560 

994540 

994519 

994499 

994479 

994459 

994438 

99W18 
9.994397 

994377 

994357 

994336 

994316 

994295 

994274 

994254 

994233 

991212 
,994191 

994171 

994150 

994129 

994108 

994087 

994066 

994045 

994024 

994003 
,993981 

993960 

993939 

993918 

993896 

993875 

993854 

993832 

993811 

993789 
9.993768 

993746 

993725 

993703 

993681 

993660 

993638 

993616 

993594 

9935-2 

993550 

99^528 

99^506 

993484 

993462 

9.J3440 

993418 

993396 

993374 

993351 


Sine. 


3.3 
3.3 
3.3 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.5 
3.5 
3.6 
3.5 
3.6 
3.5 
3.6 
3.6 
3.6 
3.5 
3.6 
3.6 
3.5 
3.5 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.7 


Tauu 


9.199713 

20)529 
201345 
202159 
20-2971 
203782 
204592 
205400 
206207 
207013 
207817 

9.208619 
209420 
210220 
211018 
211815 
212611 
213405 
214198 
214989 
215780 

9.216568 
217356 
218142 
218926 
219710 
220492 
221272 
222052 
222830 
223ti06 

9.224382 
225156 
225929 
226700 
227471 
228239 
229007 
229773 
230539 
231302 

9.232066 
232826 
233586 
234345 
235103 
235859 
236614 
237368 
238120 
238872 

9.239622 
240371 
241118 
241866 
242610 
243354 
244097 
244839 
245579 
246319 
Cotiing. 


D.  10' 


136 
136 
136 
135 
135 
136 
135 
134 
134 
134 
134 
133 
133 
133 
133 
133 
132 
132 
132 
132 
131 
131 
131 
131 
130 
130 
130 
130 
130 
129 
129 
129 
129 
129 
128 
128 
128 
128 
127 
127 
127 
127 
127 
126 
126 
126 
126 
126 
125 
126 
126 
126 
126 
124 
124 
124 
124 
124 
123 
123 


Cotang.  !;N.  sine.  N.  CO.X 


10.800287 
799471 
798655 
797841 
797029 
796218 
795408 
794(i00 
793793 
792987 
792183 

10.791381 
790580 
789780 
788982 
788185 
787389 
786595 
785802 
735011 
784220 

10.783432 
782«)44 
7818.58 
781074 
780290 
779508 
778728 
777948 
777170 
776394 

10.775618 
774844 
774071 
773300 
772529 
771761 
770993 
770227 
769461 
7(>8698 

10.767936 
767174 
766414 
766656 
764897 
764141 
763386 
762632 
761880 
761128 

10.760378 
759629 
758882 
758135 
767390 
756646 
755903 
755161 
754421 
753681 
Tang. 


15643 

15672 

15701 

15730 

15758 

I  15787 

! 15810 

15845 

i 15873 

i 15902 

15931 

I  15959 

15988 

16017 

; 16046 

1 16074 

I  16103 

116132 

i 16160 


98769 
98764 
98760 
98755 
98751 
9874() 
98741 
)8737 
98732 
98728 
98723 
98718 
98714 
98709 


98704  46 


98700 

98695 

9H690 

98686 

116189  98681 

!  16218;98676 

i  16246198671 

16275 '98667 

16304  98()62 

16333  98667 

;  16361  9H652 

;  16390  98648 

116419  98643 

;  16447i98638 

16476198633 

116505  98629 

16533!98624 

16562  98619 

16591198614 

1662098609 

l()648i98604 

16677198600 

16706i98595 

16734  ""590122 

16763:.48o85  21 

16792  98580  20 

116820  98576  19 

16849  98570 

16878  98565 

16906  98561 

1 16935  98556 

i  16964  98551 

,16992  98546 

11702198541 

I  17050  98536 

1117078  98531 

{17107  98526 

ii  17136198521 

i  1716498516 

'ini93|9S5ll 

17222  98606 

!l7250'98501 

,!  17279198496 

17308:98491 

i  17336198486 

1 17365198481 

Ij  N.  cos.lN.sine. 


80  Degrees. 


TABLE  11. 


Log.  Sinefl  and  Tangents.    (10°)    Natarnl  Sines. 


31 


0 

1 

3 

4 

5 

G 

7 

8 

9 
10 
II 
1-2 
13 
14 
16 
Iti 
17 
18 

ly 

20 

21 

22 

23 

24 

25 

2t) 

27 

28 

29 

30 

31 

32 

33 

34 

3o 

3u 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

4!» 

60 

51  1.9 

62 

53 

54 

65 

o(i 

57 

58 

59 

60 


Sine. 

.239670 
240386 
241101 
241814 
242526 
243237 
243947 
2-W(i5l) 
245363 
246069 
246775 
247478 
248181 
218H83 
249.JH3 
250JH2 
250980 
251677 
252373 
2530o7 
253761 

.254453 
255144 
255ri34 
251)523 
257211 
25  7898 
25h583 
2592()8 
259951 
260t.33 

.261314 
261994 
262(i73 
263351 
264027 
264703 
265377 
26t>051 
266723 
267395 

. 268065 
268734 
269402 
2i00t)9 
270.35 
271400 
272064 
272726 
273388 
274049 

.374i08 
276367 
276024 
276681 
27733  7 
277991 
278644 
279297 
279948 
280599 

OMioe. 


ir|  Cosine. 


119 
119 
119 
119 
118 
118 
118 
118 
118 
117 
117 
117 
117 
117 
116 
116 
116 
116 
116 
116 
115 
115 
115 
115 
115 
114 
114 
114 
114 
114 
113 
113 
113 
113 
113 
113 
112 
112 
112 
112 
112 
HI 
111 
111 
111 
HI 
111 
110 
HO 
110 
110 
110 
110 
109 
109 
109 
109 
109 
109 
108 


9.993351 
993329 
993307 
993285 
993262 
993240 
993217 
993195 
993172 
993149 
993127 

9.993104 
993081 
993U59 
99303t) 
993013 
992990 
992967 
992944 
992921 
992898 
.992875 
992852 
992829 
992806 
992783 
992759 
992736 
992713 
992690 
992666 
.992')43 
992619 
992596 
992572 
992549 
992525 
992501 
992478 
992454 
992430 
.992406 
992382 
992359 
992335 
992311 
992287 
992263 
-992239 
992214 
992190 

9.992166 
992142 
992117 
992093 
992069 
992044 
9920.20 
991996 
991971 
991947 


Sine. 


D.  10" 


3.7 

3.7 
3.7 
3.7 
3.7 
3.7 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 


3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.1 
4.1 
4.1 
4.1 
4.1 
4.1 
4.1 


9. 


Tang. 

,216319 
247057 
217794 
218630 
249264 
249998 
250730 
261461 
252191 
252920 
263648 

.254374 
256100 
265824 
256547 
257269 
257990 
258710 
259429 
260146 
260863 

.261578 
262292 
263005 
263717 
264428 
265138 
265847 
26«)555 
267261 
267967 
2<i8671 
269375 
270077 
270779 
271479 
272178 
272876 
273573 
274269 
274964 
275658 
276351 
277043 
277734 
278424 
279113 
279801 
2«0488 
281174 
281858 
282542 
283225 
283907 
284588 
285268 
285947 
286<J24 
287301 
287977 
288652 


Co  tang. 
79  Degree! 


D.  10"'  Cotang.  iiN.(«ine.|N.  oot:. 


123 
123 
123 
122 
122 
122 
122 
122 
121 
121 
121 
121 
121 
120 
120 
120 
120 
120 
120 
119 
119 
119 
119 
119 
118 
118 
118 
118 
118 
118 
117 
117 
117 
117 
117 
116 
116 
116 
116 
116 
116 
115 
115 
116 
116 
115 
116 
114 
114 
114 
114 
114 
114 
113 
113 
113 
113 
113 
113 
112 


10.753681 
752943 
752206 
761470 
760736 
760002 
749270 
748539 
747809 
747080 
746352 

10.745(i26 
744900 
744176 
743453 
742731 
742010 
741290 
740671 
739854 
739137 

10.738422 
737708 
736995 
736283 
735572 
7348(>2 
734153 
733445 
732739 
732033 

10.731329 
730625 
729923 
729221 
728621 
727822 
727124 
726427 
726731 
725036 

10.724342 
723649 
722957 
722266 
721576 
720887 
720199 
719612 
718826 
718142 

10.717458 
716775 
716093 
716412 
714732 
714053 
713376 
712699 
712(^23 
711348 


I  17366'9K481 
P393  98476 
17422  98471 
174 

II 174 

I;  ITP 


18138  98341 

18166  98336 

18195  98331 

1822498325 

18252i98320 

18281 198315 

1830998310 

18338  98304 

18367  98299 

18396198294 

18424i982fe8 

18452198283 

1848198277 

18509|98272 

1 1 18638:98267 

1856798261 

18595  98256 

18(124  98250 

18652  98245 

1868 1198240 

1871098234 

18738  98229 

!  18767  98223 

:  18795  98218 

!  1882498212 

1 18852  98207 

[18881198201 

118910  98196 

118938,98190 

!  18967198185 

1 18995:98179 

j  19024  98174 


Tang. 


19052 
19081 


98168 
98163 


N.  COP.  N.fine. 


32 


Log,  Sines  and  Tangents.    (11°)     Natural  Sines. 


TABLK  II. 


0 

I 

2 
3 
4 

5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
2; 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 

3e 

39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
61 
52 
63 
54 
55 
56 
57 
68 
59 
60 


Sine. 

.280599 
281248 
281897 
282644 
283190 
283836 
284480 
285124 
285/66 
286408 
287048 

.287687\ 
288326 
288964 
289600 
290236 
290870 
291504 
292137 
292768 
293399 

.294029 
294658 
295286 
295913 
290539 
297164 
297788 
298412 
299034 
299«55 

.300276 
300895 
301514 
302132 
302748 
303364 
303979 
304593 
305207 
305819 

.306430 
307041 
307650 
308259 
308867 
309474 
310080 
310685 
311289 
311893 

.312495 
313097 
313698 
314-.;97 
31489/ 
315495 
316092 
316689 
317284 
317879 

Cosine.  ' 


D.  W 

lOS 
108 
108 
108 
108 
107 
107 
107 
107 
107 
107 
106 
106 
106 
106 
106 
106 
105 
105 
105 
106 
105 
105 
104 
104 
104 
104 
104 
104 
104 
103 
103 
103 
103 
103 
103 
102 
102 
102 
102 
102 
102 
102 
101 
101 
101 
101 
101 
101 
100 
100 
100 
100 
100 
100 
100 
100 
99 
99 
99 


U.S.  UP. 

.991947 
991922 
9;' 1897 
991873 
991848 
991823 
991799 
991774 
991749 
991/24 
991699 

1.991674 
991649 
991624 
991599 
991574 
991549 
991524 
991498 
991473 
991448 

1.991422 
991397 
991372 
991346 
991321 
991295 
9912/0 
991244 
991218 
991193 

1.991167 
991141 
991115 
991090 
991064 
991038 
991012 
990986 
990960 
990934 

'  990908 
990882 
990855 
990829 
990803 
99077  7 
990750 
990724 
990697 
990t)71 

1.990644 
990ol8 
990591 
990565 
990538 

-990511 
990485 
990458 
990431 
990404 

~Sine. 


P.  I./' 

4.1 
4.1 
4.1 
4.1 
4.1 
4.1 
4.1 
4.2 
4.2 


.2 
.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.3 
4.3 
4.3 
4.3 


4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4,4 
4.4 
4.5 
4.5 
4.6 
4.6 


Tan-.   |D.  lu 


.288662 
2o9326 
2^9999 
290(i71 
291342 
292013 
292{)82 
293350 
294017 
294684 
295349 

i.  296013 
296677 
297339 
298001 
298662 
299322 
299980 


112 
112 
112 
112 
112 
111 
111 
111 
111 
111 
111 
111 
110 
110 
110 
i  110 
110 
110 


3006381  ,,„> 
301295  ^"-^ 
301951 


302607 
303261 
303914 
304567 
305218 
305869 
306519 
307168 
30/815 
308463 
309109 
309754 
310398 
3110-42 
311685 
31232/ 
312967 
313608 
314247 
314886 
9.315523 
316159 
316796 
317430 
318064 
318o97 
319329 
319961 
320592 
321222 
,321861 
322479 
323106 
323  733 
324368 
324983 
325607 
326231 
326»53 
327476 


109 
109 
109 
109 
109 
109 
108 
i  108 
I  108 
108 
108 
108 
i  107 
107 
107 
107 
10/ 
107 
107 
106 
106 
106 
106 
106 
106 
106 
106 
105 
106 
106 
105 
106 
106 
104 
104 
104 
104 
104 
104 
104 
104 


Cotang. 
I)egr.-es. 


10.711348 
710674 
710UJ1 
709329 
708668 
707987 
70/318 
706650 
705983 
705316 
704051 

10.70^9'<7 
703323 
702661 
701999 
701338 
700678 
700020 
699362 
698705 
698049 

10-697393 
696739 
696086 
695433 
694/82 
694131 
693481 
692832 
692185 
691537 

10-690891 
690246 
689602 
688958 
688315 
687673 
687033 
686392 
686753 
685115 

10-684477 
683841 
683206 
682570 
681936 
681303 
680u71 
680039 
679408 
678/78 
0.678149 
677621 
676894 
676267 
675642 
675017 
674393 
673769 
673147 
672526 


119081 

|il910}:« 

j;iJl38 

!:  19167 

!il9195 

119224 

1 19252 

,192U 

'l93tt9 

i 1933b 

!  1936t) 

, 19395 

i 19423 

■  19462 

'19481 

19509 

19538 

1 9566 

19595 

19623 

19652 

19680 

19/09 

19737 

19766 

197^4 

; 19823 

19851 

198b0 

19908 

1 1993/ 

' 19965 

19994 

' 20022 

20051 

200/9 

I  20108 

I  20136 

2U165 

20193 

'■20222 

,20250 

1202/9 

2030/ 

\   20336 

20364 

20393 

;  1^0421 

I v0460 

20478 

2050/ 

1 20535 

205()3 

J  20592 

i|20u20 

i!  20649 

:2(r;77 

20.06 
20734 
:20,63 
20791 


98163 

I981B7! 

^J8l52 

9«146' 

98140 

98135  1 

9812i'j 

981241 

98118! 

..(8112  I 

.^>8107j 

98101 

98096 

98090 

98084 

98079 

98073 ' 

1)8067  I 

98061  I 

98056 ' 

98050 

98044  i 

98039 \ 

98033  I 

98027  I 

98021  I 

98016 ' 

98010: 

98004  j 

97998  1 

97992; 

9  7987! 

9/981 i 

97975  1 

97969  I 

97963 

97958  i 

97962  I 

97946  j 

9.940' 

9/ 934  I 

97928 ! 

97922 : 

97916, 

97910; 

9/905 

978991 

978i;3; 

|9,887: 

9/8bll 

978761 

97809' 

97863, 

97857! 

9/851! 

97845 

9.839 

97b3ci 

9/827 

97821 

97815 


Tang. 


N.  C03.  X.t^i 


60 

59 

5b 

57 

6(i 

55 

64 

53 

62 

51 

:( 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

'34 

33 

I  32 

I  31 

30 

129 

i28 

I  27 

26 

126 

124 

i23 

,22 

'21 

I  20 

!l9 

18 

,17 

i  16 

15 

I  14 

;13 
:  -2 

\ 
10 

1 


TABLE  IJ 


Log.  Sines  and  Tangpote.    (12°)    Natural  Sines. 


3J 


Sine. 

.3178.!) 
318473 
al9»), 
31}>658 
3-20249 
3-2.JS  40 
3-214:}0 
3-2-2Uiy 
3-2-2607 
3-23194 
3-2;1780 
3-24361) 
32495U 
3-25534 
3-2iill7 
3-2()700 
3-27-281 
32  7H()'2 
3-2844-2 
3-29.J21 
3-29599 

9.3301 7t) 
330/53 
3313-29 
331903 
3324/8 
333051 
333624 
334195 
3347(i(i 
335337 

9. 33590 J 
33G475 
33/043 
3376l0 
33817b 
33874-2 
339306 
3398 ;i 
340434 
340;).>»(i 

9.341658 
342119 
342679 
343239 
34379; 
S44355 
344912 
345469 
346024 
34()579 

i^. 347134 
347687 
348240 
348792 
349343 
349893 
3  0443 
3  ).rJ92 
3=^1540 
352088 
Oosinu. 


D.  10' 


J9.0 
9S.8 
98.7 
98.6 
98.4 
98.3 
98.2 
98.0 
97.9 
97.7 
97.6 
97.5 
97.3 
97.2 
97.0 
96.9 
96.8 
96.6 
96.5 
96.4 
9ii.2 
96.1 
96.0 
95.8 
95.7 
95.6 
95.4 
95.3 
95.2 
95.0 
94.9 
94.8 
94.6 
94.5 
94.4 
94.3 
94.1 
94.0 
93.9 
93.7 
93.6 
93.5 
93.4 
93.2 
93.1 
93.0 
92.9 
92.7 
92.6 
92.6 
92.4 
92.2 
92.1 
92.0 
91.9 
91.7 
91  .t) 
91.6 
91.4 
91.3 


Cosine. 

1.990404 
990378 
99035 1 
99J324 
99)297 
9;i02;0 
9i)J243 
990215 
990188 
99)161 
990134 

1.990107 
990079 
990052 
990025 
989997 
989970 
989942 
98)J15 
989887 
9898a0 

1.989832 
989804 
989777 
989749 
989/21 
989J93 
989665 
989(i37 
989609 
989582 

.989553 
989526 
989497 
989469 
989441 
989413 
989384 
989356 
98932» 
989300 

.989271 
989243 
989214 
989186 
989167 
989128 
989100 
989071 
989042 
989014 

.988986 
988956 
988927 
988898 
988869 
988840 
988811 
988782 
988763 
988724 
Sine. 


D.  10' 


4.5 
4.5 
4.5 
4.5 
4.5 
4.5 
4.5 
4.5 
4.5 
4.6 
4.5 
4.6 
4.6 


4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.8 

4.8 

4.8 

4.8 

4.8 

4.8 

4.8 

4.8 

4.8 

4.8 

4.8 

4.9 

4.9 

4.9 


Tang. 

1.327474 
3-28aJo 
8M8715 
329334 
3-29953 
330570 
331187 
331803 
332418 
333033 
333646 

>,  334-259 
334871 
335482 
3360)3 
336702 
337311 
337919 
33852/ 
339133 
339739 

>.  340344 
340948 
341662 
342155 
342757 
343358 
343958 
344558 
345157 
345755 

1.346353 
346949 
347545 
348141 
348735 
349329 
349922 
360514 
351106 
361697 

1.352287 
362876 
353465 
354053 
354640 
355327 
355813 
356398 
356982 
367666 

1.368149 
358731 
359313 
359893 
3uOl74 
361053 
361632 
362210 
362787 
363364 

"CotHng. 


D.J_0" 

103 

103 

103 

103 

103 

103 

103 

102 

102 

102 

102 

102 

l.'>5 

102 

102 

101 

101 

101 

101 

101 

101 

101 

101 

100 

100 

100 

100 

100 

100 

100 

100 
99.4 
99.3 
99.2 
99.1 
99.0 
98.8 
98.7 
98.6 
98.6 
98.3 
98.2 
98 

98.0 
97.9 
97.7 
97.6 
97.5 
97.4 
97.3 
97.1 
97.0 
96.9 
96.8 
96.7 
96.6 
96 
96.3 
96.2 
96. 


CoLaiiif.      N.  siiK'.  N.  co>s. 


10 


10.65 


10 


lU 


10 


6726-26 
671905 
671-285 
670666 
670047 
6()9430 
668813 
668197 
667582 
66(>9()7 
66()354 
6(i5741 
6651-29 
664518 
663907 
663298 
662689 
662081 
661473 
660867 
660'2(;i 
659656 
659052 
658448 
657845 
657243 
666642 
66()042 
655442 
654843 
664245 
653(>47 
653051 
652455 
651859 
661-265 
650671 
650078 
649486 
648894 
648303 
647713 
647124 
64{)535 
646947 
646360 
644773 
644187 
643b02 
643018 
642434 
641851 
641269 
640687 
640107 
639526 
638947 
6383(>8 
637790 
637213 
636636 


20791  97815 
208-20  97809 
20848  97803 
20877  97797 
20905  97791 
20933  9v784 
20962  97778 
20990  97772 
21019  97766 
21047  97760 
21076  97754 
21104  9774- 
21132  97742 
ilH^I  97736 
21109  97729,46 
21218  97723145 


21246  97717 
21-275  97711 
21303  97705 
2133197698 
,21360  97692 
21388  97(i86 
21417  97680 


21445  97673  137 
21474  97667 
21602  97661 
21530  97656 
21559  97648 
21587  97642 
21616  97636 
21644  97630 
21672  97623 
2170197617 
21729  97611 
21768  97604 
21 78<)  97698 
2181491592 
21843  97686 
2187197679 
21899  97673 
21928  97666 
21956  97560 
21985  97653 
32013  97547 
2204197641 
22070  97534 
22098  97528 
221-26  9,521 
22165  97615 
22183  97508 
22212  97602 
22240  9,496 
22268  9  ;4«9 
22297  97483 
22325  97476 
22363  97470 
22382  95463 
22410  97467 
22438  97450 
22467  9?  444 
224'J597437 

1: 


Tang.    N.  cos.:X.8iDe. 


77  Degrees. 


34 


Log.  Sines  and  TanKents.    (13°)    Natural  Sines. 


TABLE  n. 


0 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
i: 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 


Sine. 

.352088 
352635 
353181 
353726 
354271 
354815 
355358 
355901 
356443 
356984 
35  7524 

.  3580t)4 
358603 
35.^:41 
36^678 
360216 
360752 
3612H7 
361K22 
362j:>6 
3b-.^889 

.363422 
363954 
364486 
365016 
365546 
366075 
366604 
367131 
367659 
368185 

.368711 
369236 
369761 
370286 
370808 
371330 
371852 
372373 
372894 
373414 

.373933 
374452 
3749/0 
375487 
376003 
376519 
377035 
377649 
378063 
378577 

..3790b9 
379601 
380113 
380624 
381134 
381643 
382152 
382661 
383168 
383675 
Cosine. 


D.  10"!  Cosine. 


91.1 

91.0 

90.9 

90.8 

90.7 

90.5 

90.4 

90.3 

90.2 

90.1 

89.9!, 

89.8!' 

89.7; 

89.6  i 


89.5 
89.3 

89.2 
89.1 
89.0 
88.9 
88.8 
88.7 
88.5 
88.4 
88.3 
88.2 
88.1 
88.0 
87.9 
87.7 
87.6 
87.5 
87.4 
87.3 
87.2 
87.1 
87.0 
86.9 
86.7 
86.6 
86.5 
86.4 
86.3 
86.2 
86.1 
86.0 
85.9 
85.8 
85.7 
85.6 
85.4 
85.3 
85.2 
85.1 
85.0 
84.9 
84.8 
84.7 
84.6 
84.5 


.988724 
938695 
988666 
988636 
988607 
988578 
988548 
988619 
988489 
988460 
988430 

.988401 
988371 
988342 
988312 
988282 
988252 
988223 
988193 
988163 
988133 

.988103 
988073 
988043 
988013 
987983 
987953 
987922 
987892 
987862 
987832 

.987801 
987771 
987740 
987710 
987679 
987649 
987618 
987588 
987557 
987526 

.987496 
987466 
987434 
987403 
987372 
987341 
987310 
98727S 
987248 
987217 

.987186 
987155 
987124 
987092 
987061 
987030 
986998 
9S6967 
98(i936 
986904 


Sine. 


D.  10" 


4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
6.0 
6.0 


5.0 
6.0 
5.0 
6.0 
6.0 
6.0 
6.0 
5.0 
6.1 
6.1 
6.1 
6.1 
6.1 
5.1 
6.1 
6.1 
6.1 


6.2 
6.2 
£.2 
6.2 
5.2 
6.2 
6.2 
6.2 
5.2 
5.2 
5.2 
5.2 
6.2 
5.2 
6.2 


Tang. 

9.363364 
363940 
364515 
365090 
365664 
366237 
366810 
367382 
367953 
368624 
369094 

9.369663 
370232 
370799 
371367 
371933 
372499 
373064 
373629 
374193 
374766 

9.376319 
376881 
376442 
377003 
377563 
378122 
378681 
379239 
379797 
380354 

9.380910 
381466 
382020 
382576 
383129 
383682 
384234 
384786 
386337 
385888 

9.386438 
386987 
387536 
388084 
388631 
389178 
389724 
390270 
390816 
391360 

9.391903 
392447 
392989 
393531 
394073 
394614 
395154 
395694 
396233 
396  771 


96. 
95. 
95. 
95. 
95. 
95. 
95. 
95. 
95. 
95. 
94. 
94. 
94. 
94. 
94. 
94. 
94. 
94. 
94. 
93. 
93. 
93. 
93. 
93. 
93. 
93. 
93. 
93. 
92. 
92. 
92. 
92. 
92 
92 
92. 
92. 
93. 
92. 
!91. 
91. 
91. 
91. 
91. 
91. 
91. 
91. 
91. 
90. 
90. 
90. 
90. 
90. 
90. 
90. 
90. 
90. 
90. 
89. 
89. 
89. 


Cotiin" 


Cotang.       N.sinej'N.  «os. 


10.636636 
636060 
635485 
634910 
634336 
633763 
633190 
632618 
632047 
631476 
630906 

10.630337 
629768 
629201 
628633 
628067 
627501 
626936 
626371 
626807 
625244 

10.624681 
624119 
623558 
622997 
622437 
621878 
621319 
620761 
620203 
619646 

10.619(>90 
618534 
617980 
617425 
616871 
616318 
615766 
615214 
0*4663 
614112 

10.613562 
613013 
612464 
611916 
611369 
610822 
610276 
60y/3iJ 
60J185 
60--ii)40 

10.608097 
60^.553 
607011 
606469 
605927 
605386 
604846 
604306 
603767 
603229 


22495197437 
2252397430 
22562  97424 
2268097417 
22608  97411 
22637,97404 
22666  97398 
22693197391 
j  22722 197384 
!  22750:9/378 
22778  !iT2^» 
i22807;9736o 
i  22835 197368 
22863  97351 
;  22892 197345 
973.8 
97331 
97325 
97318 
9/311 
97304 
97298 
97291 
97^84 


j  22920 
, 22948 

22977 
' 23006 
123033 
123062 
1 2^090 

23118 

23146 
|23175!972'/8 

23203  97271 
1 2323 1197264 
123260  9-25 


i  232HH 


97261 


23316  9/244 


!  1 23346 
1 1 23373 
i  23401 
i  23429 
I  23468 
I  23486 
j  23514 
23642 
!  236/1 
1^23599 
I  j 23627 
123656 
1 1 23684 
I  23712 
'123740 


Tan.i 


23797 


97237 
97230 
9/223 
97217 
97210 
97203 
97  96 
97189 
7182 
97176 
971o9 
97162 
y7166 
97148 
97141 


23769  97134 


97127 


23825  (97]  20 
23853  97113 


23882 
23910 
23938 


9710.) 
97 100 
iJ7093 


23966  970.^6 


23995 
24023 
24051 
24079 
24108 
24131) 


24164  97037 


2419V.' 


97079 
i)to,2 
97U66 
9,06) 
9/051 
97044 


9-030 


•OS.  N.sine. 


JH  Decrees. 


Log.  Sines  and  Tangents.    (14°)    Natural  Sines. 


35 


0 

I 

2 
3 
4 
6 
6 
7 
8 
9 
10 

14 
15 
U> 
17 
18 
19 
20 
21 
22 
23 
24 

25 

2(> 

27 

28 

39 

30 

31 

3-2 

33 

34 

35 

3b 

37 

38 

39 

40 

41 

42 

43 

44 

46 

4U 

47 

48 

49 

60 

61 

62 

53 

64 

66 

6b 

57 

58 

59 

W 


iliiie. 

9.383675 
384182 
38-k«8'" 

385697 
386201 
386704 
387207 
387709 
388210 
388711 
9.385)211 
389711 
390210 
390708 
391206 
391 i 03 
392199 
392695 
393191 
393685 
394179 
394()73 
395166 
395658 
396150 
396641 
397132 
397621 
398111 
398t»00 
9.399088 
3995/5 
400062 
400549 
401035 
401620 
402005 
402489 
402972 
403455 
9.403938 
404420 
404901 
406382 
405862 
406341 
40t)820 
40/299 
407777 
40.S254 
9.408  731 
409207 
4(J9682 
410157 
410632 
411106 
411679 
412052 
412521 
412996 
Cofline. 


D-  10''  Co.sine.  D.  lU"   Tang.   D.  10"|  Cotang.  IX.sine.lN.  cos 


S4.4 
84.3 

8i.l 

8l.il 

83.  ) 

«3  i 

83  7 

83  6 

83.6 

83.4 

83.3 

83.2 

83.1 

83.0 

82.8 

82.7 

82.6 

82.5 

82.4 

82.3 

82.2 

82.1 

82.0 

81.9 

81.8 

81.7 

81.7 

81.6 

81.6 

81.4 

81.3 

81.2 

81.1 

81.0 

80.9 

80.8 

80.7 

80.6 

80.6 

80.4 

80.3 

80.2 

80.1 

80.0 

79.9 

79.8 

79.7 

7y.6 

79.5. 

79.4 

79.4 

79.3 

79.2 

79.1 

79.0 

78.9 

78. 'J 

78.7 

78.6 


9.986904 

986873 

986841 

986809 

9867  78 

98674<J 

986714 

986683 

986651 

986619 

9W6587 

9 . 986556 

986523 

986491 

986459 

986427 

986395 

986363 

986331 

986299 

986266 

9.986234 

986202 

986169 

986137 

980104 

986072 

986039 

986007 

985974 

985942 

9  985909 

985876 

985843 

985811 

985778 

985745 

9^5712 

985679 

985646 

986613 

9.9;S5580 

985547 

985514 

986480 

985447 

985414 

985380 

985347 

985314 

985280 

9.085247 

985213 

985180 

985146 

985H3 

985079 

985045 

985011 

984978 

984944 


6.2 

6.3 

6.3 

5.3 

6  3 

6.3 

6.3 

6.3 

5.3 

6.3 

6.3 

6.3 

6.3 

6.3 

6.3 

5.3 

5.3 

6.4 

5.4 

5.4 

6.4 

6.4 

5.4 

6.4 

5.4 

6.4 

6.4 

5.4 

6.4 

5.4 

5.4 

5.6 

6.5 

6.6 

6.6 

5,5 

5.5 

5.5 

5.5 

5.5 

6.5 

5.5 

5.6 

5.5 

5.5 

5.5 

5.6 

5.6 

5.6 

6.6 

5.6 

6.6 

6.6 

6.6 

6.6 

6.6 

5.6 

5,6 

6.6 

5.6 


9.396771 
397309 
397846 
398383 
398919 
399455 
399990 
400624 
401058 
401591  ,  „y  ^ 
40212412?'^ 
9.402656 
403187 
403718 
404249 
404778 
406308 
405836 
406364 
406892 
407419 
9.407945 
408471 
408997 
409521 
410045 
410569 
411092 
411615 
412137 
412658 
413179 
413699 
414219  I 
414738 
415267 
415  7  75 
416293 
416810 
417326 
41 7842 
9.418358 
418873 
419387 
419901 
420415 
420927 
421440 
421952 
422463 
422974 
9.423484 
423993 
424503 
42601 1 
425519 
426(/27 
426634 
427041 
427647 
428062 

Cotang. 


89.6 
89.6 
89.5 
89.4 
89.3 
89.2 
89.1 
89.0 
88.9 


88.7 
188.6 
188.6 
'88.4 
88.3 
88.2 
88.1 
88.0 
87.9 
87.8 
87.7 
87.6 
87.6 
87,4 
87.4 
87.3 
87.2 
87.1 
87.0 
86.9 
86.8 
86.7 
86.6 
86.5 
86.4 
86.4 
86.3 
86.2 
86.1 
86.0 
86.9 
86.8 
86.7 
86.6 
86.5 
86.5 
86.4 
86.3 
86.2 
86.1 
85.0 
84.9 
84.8 
84.8 
84.7 
84.6 
84.6 
84.4 
84.3 
84.3 


10.603229 
602(i91 
602154 
601617 
601081 
600645 
600010 
599476 
698942 
598409 
597876 
10.597344 
696813 
696282 
695751 
595222 
594692 
594164 
693636 
593108 
592581 
10.592055 
691529 
691003 
690479 
589965 
589431 
688908 
588385 
587863 
587342 
10.586821 
686301 
685781 
585262 
684743 
584225 
583707 
583190 
682674 
582158 
10.581642. 
681127 
680613 
580099 
679586 
679073 
678560 
578048 
677637 
677026 
10.676516 
576007 
575497 
674989 
574481 
673973 
573466 
572959 
572453 
571948 


24192  97030,60 

24220,97023  I  69 
24249,97015168 
24277  97008.67 
1 2430o'^7001  '  66 
24333'9(>994 !  55 
243(i2!f)6987  [  64 
24390J  96980  53 
24418  96973;  52 
24446^96966  1  61 
24474  96959  |  60 
24503 196952 ,  49 


24531196945 
24559 196937 


i  24587 
i24<il5 
' 24644 
124672 
24700 
24728 


96930 
96923 
96916 
9<>909 
96902 
96894 


24756  96887 
24784196880 1  39 
24813  96873  38 


124841 
124869 

2489 
! 24925 
I  24954 
: 24982 
126010 
; 25038 
i  250i)6 
1 25094 
125122 
125151 

25179 
125207 
i  25235 
1 25263 
'25291 
, 25320 


9()866  i  37 

9<)858  I  36 

96851  I  35 

96844 

96837 

96829 

96822 

96815 

96807 

96800 

96793 

96786 

>6778 

96771 

96764 

96766 

96749 

96742 


25348196734 


125376 
25404 
26432 
125460 
25488 
125516 
i 25545 
25573 
25<>01 
25629 
i  25667 
25685 
25713 
25741 
25766 
25798 
258- Ui 
25854 


96727 
96719 
9(J712 
96705 
96697 
96690 


34 

33 

32  I 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 


96682 1  1 


25S8' 


96670 
96667 
96660 
96653 
96645 
9()638 
96t)30 
96623 
96615 
96t)08 
96600 
J6593 


Tang. 


N.  cofuN.mnH. 


75  liegrttes 


36 


Log.  Sines  and  Tangents.     (15*^)     Natural  ^mes. 


TAULE  II. 


0 

I 
2 
3 
4 
6 
6 
7 
8 
9 
10 
U 
1-2 
13 
14 
15 
i6 
17 
18 
19 
20 
21 
22 
23 
24 
25 
2ti 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
60 
61 
52 
53 
64 
i  55 
56 
57 
68 
59 
60 


Sine. 


9. 41 299 J 
413467 
41393S 
414408 
4148/8 
415347 
415815 
416283 
416751 
417217 
41 ;684 

9.418150 
418615 
419079 
419544 
420007 
420470 
420933 
421395 
421857 
422318 

9.4227  78 
423238 
4236.^7 
424166 
s24615 
425073 
425630 
425987 
426443 
426899 

9.427li54 
427809 
428263 
428717 
429170 
429623 
430076 
430627 
430978 
431429 

9.4318/9 
432329 
432778 
43b226 
433676 
434122 
434569 
435016 
435462 
436908 
.436353 
436798 
437242 
43  7686 
438129 
438572 
439014 
439456 
439897 
440338 
Cosine. 


D.  W 


78.5 
78.4 
78.3 
78.3 

7S  <: 

78  1 
78.0 
77.9 
77.8 
77.7 
77.6 
77.5 
77.4 
77.3 
77.3 
77.2 
77.1 
77.0 
76.9 
75.8 
76.7 
76.7 
76.6 
76.6 
76.4 
76.3 
76.2 
76.1 
76.0 
76.0 
75.9 
75.8 
75.7 
75.6 
76.6 
76.4 
76.3 
75.2 
76.2 
76.1 
76.0 
74.9 
74.9 
74.8 
74.7 
74  6 
74.6 
74.4 
74.4 
74.3 
74.2 
174.1 
174.0 
174.0 
73. 9 
73.8 
73.7 
73.6 
73  6 
73.5 


Uasiiic. 

9  984944 
984910 
984876 
984842 
984808 
984774 
984740 
984706 
984ri72 
984637 
984603 

9.984569 
984535 
984500 
-984466 
984432 
984397 
984363 
984328 
984294 
984259 

9.984224 
984190 
984155 
984120 
984085 
984050 
984U15 
983981 
983946 
983911 

9.983875 
983840 
983805 
983770 
9837S6 
983 < 00 
983664 
983629 
983594 
983558 

9.983523 
983487 
983462 
983416 
983381 
983346 
983309 
983273 
983238 
983202 
983166 
983130 
983094 
983058 
983022 
982986 
982960 
S82914 
982878 
982842 


5.7 
5.7 
5.7 
5.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
5.7 
5.7 
6.8 
5.8 


D.  lU"   Tang. 

9.428052 

428557 
429062 
42966;> 
4300  7 IJ 
430573 
431075 
431677 
432079 
432580 
433080 

9.433580 
434080 
434579 
435078 
435576 
436073 
436670 
437067 
437563 
438059 

9.438554 
439048 
439543 
440036 
440529 
441022 
441614 
442006 
442497 
442988 

9.443479 
143968 
444458 
444947 
445435 
445923 
446411 
446898 
447384 
447870 
.448356 
448841 
449326 
449810 
460294 
450777 
451260 
451743 
462225 
452706 

9.453187 
453668 
454148 
454628 
455107 
466586 
456064 
456542 
457019 
467496 


6.8 
6.8 
5.8 
5.8 
6.8 
6.8 
6.8 
6.8 
5.8 
5.8 
5.8 
6.8 
6.8 
5.8 
5.8 
6.9 
5.9 
5.9 
6.9 
6.9 
5.9 


6.9 
5.9 
6.9 
5.9 
5.9 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 


D.  10^' 

84  2 

84 

84  . 

h3.9 

83.8 

83.8 

83.7 

83.6 

83.6 

83.4 

83.3 

83.2 

83.2 

83.1 

83.0 

82.9 

82.8 

82.8 

82.7 

82.6 

82.6 

82.4 

82.3 

82.3 

82.2 

82.1 

82.0 

81.9 

81.9 

81.8 

81.7 

81.6 

81.6 

81.5 

81.4 

81.3 

81.2 

81.2 

8i.l 

81.0 

80.9 

80.9 

80.8 

80.7 

80.6 

80.6 

80.6 

80.4 

80.3 

80.2 

80.2 

80.1 

80.0 

79.9 

79.9 

79.8 

79.7 

79.6 

79.6 

79.5 


25882 
2591 J 
2593o 
2596c> 
25994 


26079 
2610/ 
26136 
26163 
26191 
26219 
26247 
26276 


26331 
26369 
26387 
26415 


Cotang. 
li  Degrees. 


cotaug.       N.  siu. 

10.571948 

671443 

670938 

670434 

669930 

6t)9427 

5ii8925 

5(i8423 

567921 

567420 

666920 
10.566420 

6()5920 

61)6421 

664922 

664424 

6(i3927 

5b3430 

56-2933 

662437 

561941 
10.661446 

660952 

660457 

559964 

559471 

568978 

658486 

657994 

567503 

557012 
10.556521 

556032 

655542 

555053 

554565 

654077 

&53589 

553102 

552616 

652130 
10.561644 

551159 

550674 

650190 

549706 

549223 

648740 

548257 

6477 i 6 

6472§4 
10.646813 

646332 

645852 

645372 

544893 

644414 

54393G 

543458 

642981 

542504 


96593 
yiioH.") 
9657-. 
96570 
96662 


26022196566 
26050|9()547 


96540 
96632 
96624 
96517 
96509 
96502 
96494 
96486 


2630Ji  96479 


96471 
96463 
9()456 
964-18 


!!  26443  96440 


26471 
26500 
26528 
26556 
26584 
26612 
26640 
26668 
26696 
26724 
26/52 
26780 
2680b 
26836 
26864 


96433 
96425 
96417 
96410 
96402 
96394 
06386 
96379 
96371 
96363 
96355 
96347 
96340 
96332 
96324 


26892  96316 


26920 


26948  96301 


126976 
i  27004 

I  27032 
j 2/060 
! 27088 
,27116 
127144 
; 27172 
27200 
1 27228 
■27256 
,127284 
127312 


27340  96190 


2736b 


i  27396  96174 
127424  96166 


9630b 


96293 
96285 
96277 
96269 
96261 
96-253 
96241) 
96238 
96230 
96-.i22 
96214 
96206 
96198 


96lb2 


27452 '96158 


j 27480 
127608 
127  536 


127564  96126 


96150 
ybl42 
96134 


Tang.   !]  N.  cos.|X.sine.  ' 


Log.  Sinus  and  Tangents.    (16°)    Katural  Sines. 


37 


D.  10^' 

73.4 
73.3 
73.2 
73-1 
73.1 
73.0 
72.9 
72.8 
72.7 
72.7 
72. G 
72.6 
;2.4 
72.3 
72.3 
72.2 
72.1 
72. U 
72.  U 
71.9 
71. « 
71.7 
71.6 
71. (i 
71.5 
71.4 
71.3 
71.3 
71.2 
71.1 
71.0 
71.0 
?0.9 
70.8 
70.7 
70.7 
70. G 
70.6 
70.4 
70.4 
70.3 
70.2 
70.1 
70.1 
70.0 
G9.9 
Gy.8 
G9.8 
(,9.7 
G9.G 
G9.5 
t)9.5 
G9.4 
1,9.3 
GO. 3 
09.2 
G9.1 
G9.0 
()1>.0 
G8.9 


Sim 


0 
1 

9 
3 
4 
5 

7 

8 

9 
10 
II 
12 
13 
14 
15 
IG 
17 
18 
19 
20 
21 
22 
23 
24 
25 
2G 
27 
28 
29 
30 
31 
32 
33 
34 
35 
3G 
37 
38 
39 

JO 
41 
42 
43 
44 
45 
4G 
47 
48 
49 
50 
51 
52 
53 
54 
55 
5G 
57 
58 
59 

GO 

_j 


:).4l0338 
440/^8 
441218 
44lo58 
44209(> 
442635 
442973 
443410 
443847 
444284 
4  44720 
'445165 
445590 
446025 
441)459 
44()S93 
447 32G 
4-4/769 
448191 
44SG23 
449054 
449485 
'449916 
450345 
450775 
451204 
451G32 
4620., 0 
452488 
452915 
453342 
,453708 
464194 
454ol9 
456044 
4.-64G9 
456893 
45G31G 
45G739 
457 1G2 
45; 584 

9  458U0J 
458427 
458848 
4592G8 
459G88 
4<>0108 
4G0527 
4G094G 
4()13G4 
4G1782 

!>.  402199 
4G2G1G 
4«i3032 
4G3448 
4()38G4 
4G42;9 
4G4()94 
4G5108 
4()o622 
4G5936 
Cosiii". 


Cosine. 

►  .982812 
982^05 
982;(i9 
982733 
982t)9ti 
982G(iO 
982(i24 
982587 
982651 
982514 
982477 

)  982441 

^  982404 
9823G7 
982331 
982294 
982257 
982220 
982183 
9H214G 
982109 

(1982072 
982035 
981998 
981 90 1 
981924 
9818«G 
981849 
981812 
981774 
981737 

I.981G99 
981ou2 
y81G25 
981587 
981549 
981  ,12 
9hl474 
98143G 
981399 
9bl3Gl 

).  98 1323 
981286 
981247 
9812U9 
9811/1 
981133 
981096 
981057 
981019 
y80i>hl 

). 9809 4 2 
980904 
9808GG 
980827 
980789 
98IJ750 
980712 
980o73 
9aoG35 
98069G 
Sine. 


li.O 
li.O 
(i.l 
G.l 
G.l 
G.l 
6.1 
G.l 
G.l 
6.1 
6.1 
6.1 
6.1 
6.1 
G.l 
6.1 
G.l 
6.2 
6.2 
G.2 
G.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 


9. 


6.3 
6.3 
6.3 
6.3 
G.3 
G.3 
6.3 


3 

3 

3 

3 

3 

4 

4 

4 

4 

4 

4 

6.4 

6.4 

G.4 

6.4 

6.4 

G.4 

6.4 

6.4 


'fang. 

45749G 
457953 
458449 
458925 
459400 
459875 
4G0349 
4G0823 
4612^7 
4Gli70 
402242 
462714 
4G3186 
4G3G58 
4G4129 
4(i459y 
4G50oy 
465539 
406008 
4GG476 
46G945 
467413 
467880 
4G8347 
468814 
4G9280 
4G9;4tJ 
470211 
4/0076 
471141 
471G05 
472058 
472532 
472996 
473457 
473919 
474381 
474842 
476303 
4/6/03 
476223 
476683 
477142 
477001 
478069 
478517 
478975 
479432 
479889 
480345 
480801 
481257 
481712 
482 1G7 
48202 1 
483076 
483529 
483982 
484436 
484887 
486339 


D.  10' 


Cotang. 


Cotan-, 

10.542604 
642027 
541551 
6410/5 
640G00 
540125 
539ii51 
639177 
638703 
538230 
537758 

10.537286 
636814 
53G342 
536871 
536401 
634931 
5344G1 
633992 
533524 
633055 

10.532587 
632120 
631063 
531186 
630720 
530264 
629/89 
629324 
628869 
628395 

10.527932 
527468 
627006 
526543 
626081 
525619 
526168 
524097 
524237 
623777 

10.623317 
522858 
522399 
521941 
621483 
521026 
6205G8 
5201 1 1 
519065 
519199 

10.518743 
618288 
pl7833 
517379 
616925 
516471 
516018 
5156G5 
615113 
614661 


N. 


,,X.  CO8. 


27664'9G126  60 
27692,901 18  159 
2/020  90110  68 
,27G48!9(,102|67 
27076190094   56 


2770419G086 
27731 I9G078 
27769  9(i0;0 
27787!9(,0G2'52 
27815;yo054i61 
27843|9004G:60 


i  2787 1 
I  27899 
2792/ 
! 27955 
127983 
128011 
: 28039 
|2bOG7 
, 26095 


90(»3'?  ;  49 
9G029 I  48 
90021  47 
90013  46 
9u005  46 
95997  I  44 
95989  I  43 
95981  42 
959/2  41 


28123196904  40 


28150  9695G 
:  281  i  8  95948 
!  2820(i|95940 
i28234j95931 
28202|95923 
28290J96916 
28318|95907 
:  28340196898 
28374195890  ■  31 
;  28402|958h2  I  30 
i  28429:95874  29 
2845/95805 
28486  96857 
!  28513195849 
'2854195841 


2860995832  24 


28597  96824 


':  28025 
28052 
■  28080 
; 287 08 
' 28730 


95810 
»5807 
95799 
95:91 
95782 


28704195774 
95700 


28792 
2882U 
28847 
28875 
'2890^ 
!  28931 
'289.59 


95767 
95;49jl4 
95740  I  13 
95732  j 12 
95724 
957  1 5 


28987  ■J5/ 07 


29015 
2904-^ 
290/0 
29098 
2blvJ0 
29154 
29182 
292  O.v 
29247 


Tang. 


N.  COS.  N.sinc. 


95(i98 
96090 
95()8l 
95073 
95(iG4 
95056 
95047 
951.39 
96G30 


73  Dirgrees. 


38 


Log.  Sines  and  Tangents.  (17°)  Natural  Sines. 


TABLE  IL 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 


9.465935 
466348 
466761 
467173 
467585 
467996 
4()8407 
468817 
469227 
469(>37 
470046 
9.470455 
470863 
471271 
471679 
472086 
472492 
472898 
473304 
473710 
474115 
9.474519 
474923 
475327 
475730 
476133 
476536 
476938 
477340 
477741 
478142 
9.478542 
478942 
479342 
479741 
480140 
480539 
480937 
481334 
481731 
482128 
9.482525 
482921 
483316 
483712 
484107 
484501 
484895 
485289 
485682 
486075 
1.486467 
486860 
487251 
487643 
488034 
488424 
488814 
489204 
489593 
4899H2 
Cosine 


D.  10"(  Co.sine. 


68.8 

68.8 

68.7 

68.6 

68.5 

68.5 

68.4 

68.3 

68.3 

68.2 

68.1 

68.0 

68.0 

67.9 

67.8 

67.8 

67.7 

67.6 

67,6 

67.5 

67.4 

67.4 

67.3 

67.2 

67.2 

67.1 

67.0 

66.9 

66.9 

66.8 

66.7 

66.7 

66.6 

66.5 

66.5 

66.4 

66.3 

66.3 

66.2 

66.1 

66.1 

66.0 

65.9 

65.9 

65.8 

65.7 

65.7 

65.6 

65.5 

65.5 

65.4 

66.3 

65.3  j 

65.21 

65.  l! 

65.1; 

65.0 1 

65.0 

64.9 

64.8 


.980596 
980558 
980519 
980480 
980442 
980403 
980364 
980325 
980286 
980247 
980208 
980169 
980130 
980091 
980052 
980012 
979973 
979934 
979895 
979855 
979816 

9.979776 
979737 
979697 
979658 
979618 
979579 
979539 
979499 
979459 
979420 

9.979380 
979340 
979300 
979260 
979220 
979180 
9791401 
9791001 
979059 
979019 

9.978979 
978939 
978898 
978858 
978817  I 
978777 
978736  I 
978696  I 
978655  I 
9786151 

9.978574 
978533 
978493  I 
978452 
978411 
9783701 
978329 ' 
978288 
978247 
978206 


D.  10" 


Sine. 


6.4 

6.4 

6.5 

6.5 

6.5 

6.5 

6.5 

6.5 

6.5 

6.5 

6.5 

6.5 

6.5 

6.5 

6.5 

6.5 

6.5 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.7 

6.7 

6.7 

6.7 

6.7 

6.7 

6.7 

6.7 

6.7 

6.7 

6.7 

6.7 

6.7 

6.7 

6.7 

6.8 

6.8 

6.8 

6.8 

6.8 

6.8 

6.8 

6.8 

6.8 

6.8 

6.8 

6.8 


Tang. 


9.485339 
485791 
486242 
486693 
487143 
487593 
488043 
488492 
488941 
489390 
489838 

9.490-'86 
490733 
491180 
491627 
4926 73 
492519 
49296i 
493410 
493854 
494299 

9.494743 
495186 
495630 
496073 
496515 
496957 
497399 
497841 
468282 
498722 

9.499163 
499603 
600042 
500481 
600920 
501359 
501797 
502235 
502672 
503109 

9.503646 
503982 
504418 
504854 
505289 
605724 
506159 
606593 
607027 
607460 

9.507893 
508326 
508759 
509191 
609622 
6100.34 
510485 
610916 
511346 
511776 
Co  tans'. 


f'  Oegrces. 


I).  10"|  Cotang.  I  N.  sine  N.  cos.' 
;:29237J95630 


75.3 
75.2 
75.1 
75.1 
75.0 
74.9 
74.9 
74.8 
74.7 
74.7 
74.6 
74.6 
74.5 
74.4 
74.4 
74.3 
74.3 
74.2 
74.1 
74.0 
74.0 
74.0 
73.9 
73.8 
73.7 
73.7 
73.6 
73.6 
73.6 
73.4 
73.4 
73.3 
73.3 
73.2 
73.1 
73.1 
73.0 
73.0 
72.9 
72.8 
72.8 
72.7 
72.7 
72.6 
72.5 
72.5 
72.4 
72.4 
72.3 
72.2 
2.2 
72.1 
72,1 
72.0 
71.9 
71.9 
71.8 
71.8 
71.7 
71.6 


10.514661 

614209 

613758 

513307 

512857 

612407 

611957 

511508 

511059 

510610 

510162, 

10.609714 

509267 

508820 

508373  I 

507927 ! 

507481 : 

507035 

506590 

506146 ; 

505701  I 

10.505257! 

604814 

604370 

503927 ! 

503485  i 

603043; 

6026011" 

602159  it 

501718: 

50127811 

10.600837  I 

500397' 

499958!! 

499519 L 

499080!^ 

498641 i 

498203  ij 

497765  ij 

497328  I 

496891  Ij 

10.496454!: 

496018  ji 

495582!! 

495146!; 

494/11 

494276  m 

493841 i 

493407  1 1 

49297311 

492640 

10.492107  ;i 

491674|! 

49124111 

490809;! 

4903  78;; 

489946 

489515 

489084 

488654 

488224 


29265195622 
29293  !956 13 
2932195605 
29348195596 
293  76;95588 
'29404  95579 
29432  95571 
2946095562 
29487195554 
295 15  [95545 
29543  !95536 
2957195528 
29599 '955 19 
!  29626  i955 11 
■  29654 195502 
29682 '95493 
29710  954«5 
29737195476 
29766195467 
29793I95459 
29821 195450 
29849J95441 
2987()  195433 
29904  !95424 
29932:95416 
29960195407 


2998, 
30015 
300-i3 
300/1 
30098 
30126 
30154 
30182 
30209 
3023/ 
30265 
30292 
30320 
3034b 
30376 
30403 
304^1 
30^59 


95398 
95389 
95380 
95372 
95363 
95354 
95345 
95337 
95328 
95319 
95310 
95301 
95293 
95284 
95276 
95266 
95257 
95248 


30  486  [95210 
30o  14  95231 
30542 195222 
305/0195213 
3059  /  95204 
30625|95195 


30663 
306«0 


95186 
951/7 


30 1 08  95168 
3073b  195 169 


3iJ7(Ji 
30791 
30819 
30846 
308/4 
3090- 


Tang. 


N.  CO--.  N. 


95150 
95142 
95133 
95124 
95115 
95106 


60 
69 
58 
57 
56 
55 
64 
53 
52 
51 
60 
49 
48 
47 
46 
45 
44 
43 
42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 


TABLE  II. 


Log.  Sines  awl  Tangents.    (18°)    Natural  Sines. 


39 


Tang.     |D.  lO 

.511776 
OV2206 
5l'2blio 
613064   ^ 

5i3y'2i  I;;-* 

614349  ■'^••* 
614777 
516204 
615631 
616057 
.616484 
516910 
517335 
517761 
6181«5 
618610 
619034 
519458 
51 9882 
6'20306 


Sine. 

.489982 
490371 
4W0'<59 
491147 
491535 
491922 
492308 
492t)95 
493081 
49346() 
493851 
.494236 
494^21 
495005 
495388 
4957 ?2 
496154 
496537 
496919 
497301 
497()82 
.4980<)4 
498444 
498825 
499204 
499584 
499963 
600342 
600721 
601099 
6014;6 
.501854 
602231 
60260? 
602984 
603360 
603736 
604110 
604485 
604860 
605234 
.505608 
605981 
506354 
506727 
60-099 
60/471 
60-843 
508214 
508585 
508956 
1.509326 
-  609()9() 
510065 
510434 
610803 
611172 
611540 
511907 
512275 
512642 
Cosiuf. 


p.  10" 

64.8 
'64.8 
64.7 
64.6 
64,6 
64.5 
64.4 
MA 
64.3 
64.  Q 
64.2 
64.1 
64.1 
64.0 
63.9 
63.9 
63.8 
63.7 
63.7 
63.6 
63.6 
63.6 
63.4 
63.4 
63.3 
63.2 
63.2 
63.1 
63.1 
63.0 
62,9 
62.9 
62.8 
62.8 
62,7 
62,6 
62.6 
62.5 
62.5 
62.4 
62.3 
62,3 
62.2 
62,2 
62.1 
62.0 
62.0 
61.9 
61.9 
61.8 
61.8 
61.7 
61.6 
61.6 
61,5 
61.5 
61.4 
61.3 
61.3 
61.2 


Co.sinc 


D.  1(F 


.978206 
9781»)5 
978124 
978083 
978042 
978001 
977959 
977918 
977877 
977836 
977794 

.977762 
977711 
977669 
977628 
977686 
977644 
977603 
977461 
977419 
977377 

.977335 
977293 
977251 
977209 
977167 
977125 
977083 
977041 
976999 
97()967 
976914 
976872 
976830 
976787 
976745 
976702 
976660 
976617 
976574 
976532 

.976489 
97644<> 
976404 
976361 
97ti318 
976275 
976232 
976189 
976146 
970103 
.9760()0 
97b017H 
9769/4 
975930 
976887 
975844 
975800 
976.67 
975714 
975670 


Sine. 


6.8 
6.8 
6.8 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7,0 
7.0 
7,0 
7,0 
7.0 
7,0 
7.0 
7.0 
7.0 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7,1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7,2 
7.2 
7.2 


71.6 
71.6 
71.6 


71.3 
71.2 
71.2 
71.1 
71.0 
71.0 
70.9 
70,9 
70.8 
70.8 
70.7 
70.6 
70.6 
70.5 


.620/28  r^,.'  , 
/0.4 

70.3 


621151 
521573 
621996 
622417 
522838 
623259 
623680 
624100 
624520  ^^,  „ 

.524939  r^.-r. 

625359 

625778 

626197 

626616 

627033 

627451 

627868 

628286 

62tt702 
.629119 

629635- 

629950 

630366 

630781 

631196 

631611 

532025 

6324u9 

632853 
.633266 

633(i79 

634092 

634504 

534916 

535328 

635739 

5ci6150 

636561 

5;i6972 

Ck)tan};, 

il  Degrees, 


70.3 
70.3 
70.2 
70.2 
70.1 
70.1 
70.0 


69.9 
69.8 
69.8 
69.7 
69,7 
69.6 
69.6 
69.5 
69.5 
69.4 
69,3 
69.3 
69.3 
69.2 
69.1 
69.1 
69.0 
69.0 
68.9 
68.9 
68.8 
68.8 
68.7 
68.7 
68.6 
68,6 
68.5 
68.5 
68.4 


Cotang.      N.  pine.jX.  cos.l 


10.488224 
487794 
487365 
486936 
486507 
486079 
485b51 
485223 
484796 
484369 
483943 

10.483516 
483090 
482665 
482239 
481816 
481390 
480966 
480542 
480118 
479695 

10.479272 
478S49 
478427 
478005 
477583 
477162 
476741 
476320 
475900 
475480 

10.475061 
474641 
474222 
473803 
473385 
472967 
472549 
472132 
471715 
471298 

10.470881 
470465 
470050 
469634 
4(i9219 
468804 
468389 
4679/5 
467561 
467147 

10.466734 
466321 
4b5908 
465496 
465084 
464672 
464261 
463850 
463439 
463028 

"r^ng- 


30902196106,60 
3092995097  i  69 
30957 195088  i  58 
309h6;950;9[67 
31012195070166 
31040  950<)1  155 


31068 
31095 
31123 


95052  I  54 
95043  53 
95033  62 
31151195024;  51 
3]  178  95015  50 


:  31206 
i 31233 

31261 
;  31289 
131316 

31344 


i  3 1372194052 


31399 
31427 
131454 


95006  49 
94J>97  48 
94988 { 47 
94979  I  46 
94970 1  46 
94961  I  44 


94943 
94933 
94924 
:  31482  94915 
i31.010  94906|38 
'31537194897  I  37 
,31505  94888  36 
; 3159394878  I  36 
31620  948t>9  34 
3 1648  94860  i  33 
31t)75  94851  '32 
,31703  94842131 


3173094832  30 

31758  94823  29 

31786  94814  28 

r3l813  94805|27 

31841  94795!  26 

,31868  94786125 

131896  94777  I  24 

131923  94768123 

;i  31951  94758  122 

,31979  94749121 

0  32006  94740  I  "20 

[32034  94730  19 

1132061  94i21  18 

!  32089  94712  17 

i|  321 16  94702  16 

li  32144  94693  I  15 

Ii3217194(.84!l4 

;  132 199  94674;  13 

32227194666  I  12 

32250  94(i56  i  1 1 

32282  94()46 

32309  94637 

32337 :94627 

32364'94()18 

32392:94009 

32419i94599 

32447  J94590 

32474-94580 

32502194571 

32529  94501 

3255.  94552 


N.  W8.  N'.sine. 


TABLE  II. 


Log.  Sines  and  Tangent8.    (20°)    Natural  Sineb. 


41 


SJue.  ID.  10' 


1.53405-2 
634399 
634745 
63509-2 
6354:J8 
635783 
5361-29 
53j474 
63(i8lH 
.53/l(i3 
53750? 

I.537S51 
538 194 
638538 

-638880 
539-2-23 
53951)5 
539907 
540-249 
540590 
540931 

L54l-2r'2 
541013 
541953 
642-293 
54-2(i3-2 
54-2971 
543310 
643()49 
543987 
5443-25 

I.544<)(i3 
545000 
545338 
545074 
54001 1 
54()347 
54()ti83 
647019 
547354 
647089 

1.5480-24 
648359 
548093 
5490-27 
549300 
549093 
5500-.0 
650359 
550i)9'2 
6510-24 
651350 
56108; 
55-2018 
55-2349 
55-2<>80 
653010 
553341 
5.53070 
654000 
564329 

Cosine. 


57.8 
67.7 
57.7 
57-7 
57.0 
57.6 
57.5 
57.4 
57.4 
57.3 
57.3 
57.2 
57.2 
57.1 
57.1 
57.0 
57.0 
50.9 
50.9 
50.8 
50.8 
50.7 
50.7 
50.0 
50.0 
50.5 
50.5 
60.4 
50.4 
50.3 
50.3 
50.2 
5(i .  2 
50.1 
50.1 
50.0 
50.0 
55.9 
55.9 
55.8 
65.8 
55.7 
55.7 
55.0 
55.0 
55.5 
55.6 
55.4 
55.4 
55.3 
65.3 
55.2 
65.2 
65.2 
55. 1 
55. 1 
65.0 
55.0 
54.9 
54.9 


Co.«iue. 


D.  10' 


9.9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 

9.9 
9 
9 
v9 
9 
9 
9 
9 
9 
9 

9.0 
9 
9 
9 
9 
9 
9 
9 
9 
9 

9.9 
9 
9 
9 
9 
9 
9 
9 
9 
9 

9.9 
9 
9 
9 
9 
9 
9 
9 
9 
9 

9.9 
9 
9 

y 

9 
9 
9 
9 
9 
9 


2980 

2940 

•2894 

-2H48 

-2802 

2755 

2/0  < 

2003 

•2()17 

-2570 

2524 

2478 

-2431 

2385 

2338 

■2-291 

2245 

2198 

2151 

2105 

2058 

-2011 

1904 

1917 

1870 

1823 

1770 

1729 

1082 

1035 

1588 

1540 

1493 

1440 

1398 

1351 

1303 

1-260 

1-208 

1101 

1113 

1000 

1018 

09  70 

0922 

0874 

0827 

0779 

0731 

0083 

iJ036 

0586 

0538 

0490  j 

0442! 

0094  1  y 

0345  I  **■ 
0297 
0249 
0200 
0152 
.Sin«. 


rang. 


9.501(X)0 

6til459 

601S51 

602244 

502030 

5f>3028 

503419 

50.3811 

604202 

604592 

604983 
9.5«.5373 

6ti5703 

600153 

600542 

600932 

607320 

607709 

508098 

608480 

608873 
9.509-201 

509048 

570035 

5704-22 

670809 

671195 

571581 

571907 

672352 

672738 
9.6731-23 

573607 

573892 

674270 

574000 

576044 

575427 

575810 

570193 

670570 
9.576958 

577341 

677723 

5/8104 

5/8480 

578807 

579*248 

6790-29 

5800*J9 

580389 
9.580709 

581149 

6815-28 

681907 

682280 

582W;5  I 

583043  I 

6r;31-22l":;" 

68?.800   ^'^•" 


D.  10" 


05.6 
05.4 
t)5.4 
()5  3 
05.3 
()5.3 
()5.2 
05.2 
05.1 
05.1 
05.0 
05.0 
04.9 
04.9 
04.9 
04  8 
04.8 
04.7 
04.7 
04.0 
04.0 
04.5 
04.5 
04.5 
04.4 
04.4 
04.3 
04.3 
04.2 
04.2 
04.2 
04.1 
04.1 
04.0 
04.0 
03.9 
03.9 
03.9 
03.8 
03.8 
03.7 
03.7 
63.6 
03.6 
63.6 
63.5 
03.6 
03.4 
03.4 
03.4 
03.3 
63.3 
63.2 
63.2 
63.2 
63.1 
63.1 


684177 
Co  tang. 


62.9 


Cotang.   N.  sinr.iN.  co8. 


10.438934 
438541 
438149 
437750 
437364 
430972 
430581 
430189 
436798 
436408 
435017 

10.434027 
434237 
433847 
433468 
433008 
432080 
432-291 
431902 
431514 
431127 

10.430739 
430352 
4299o5 
429578 
4-29191 
428805 
428419 
428033 
427048 
427-202 

10.426877 
426493 
4-26108 
4257-24 
425340 
424950 
4-24673 
424190 
423807 
4234-24 

10.4-23041 
422669 
-42*2277 
421896 
421614 
421133 
420752 
420371 
419991 
419011 

10.419*231 
418851 
418472 
,  418093 
4^17714 
417335 
410957 
410578 
410200 
4158-23 

Tang.   ' 


34202|939()9  1 60 

34229;93959  69 

34-257(93949 

34-284193939 

343111939*29 

343^9193919 


34360 
34393 
344-21 
34418 
34475 
34503 
34530 
34557 
34584 
3401-2 
34039 
34000 
34094 
34721 
34748 


93909 
93899 
93889 
93879 
93809 
93859 
93849 
93839 
93829 
93819 
i>380y 
93799 
93789 
93779 
93709 


34775  93759 
34803193748 
3483093738 
34857  93728 
34884  93718 
34912|93708 
34939l93«i98 
3490o!93()88 


34993 
35021 
35048 
350-5 
35102 
36130 
3515< 
35184 
35211 
35239 
35-200 
35293 
36320 
3534/ 
35375 
3540^ 
3545^9 
35450 
35484 


930/  i 
930t)7 
93057 
93()47 
93037 
93(i20 
93010 
93000 
93590 
9o5o5 
33675 
93506 
93665 
9o644 
93534 
935*24 
93614  , 
93603  i  14 
93493 j  13 
35511193483'  12 
1 36538193472  '  1 1 
36505I93402  '<  1 D 
13569*^193452 


35019 

3504 

35()V4 

36701 

367*28 

35756 

3578J 

35810 

3583 


I  N.  COS.  N.sinf 


93441 

93431 

^J34*20 

93410 

93400  I 

93.*i89  I 

93379 

933t)8 

93368 


69  Degrees. 


42 


^og.  Sinea  and  Tangents.    (21°)    Natural  Sines. 


TABLE  IE. 


Sine. 

9.554329 
654658 
654987 
6553 1 5 
655(>43 
555971 
656-299 
556626 
556953 
557280 
557606 
557932 
558258 
558583 
658909 
659234 
65y55S 
659883 
56020/ 
560531 
560855 

9.561178 
661501 
561824 
562146 
562468 
562790 
563112 
563433 
663755 
564075 
56439() 
564716 
565036 
665356 
565676 
665995 
566314 
566632 
666951 
667269 
567587 
567904 
568222 
568539 
668856 
669172 
569488 
5()9804 
570120 
670435 

9.570751 
571066 
571380 
571695 
672009 
572323 
572636 
572950 
573263 
673575 
Cosine. 


D.  10' 


54.8 
54.8 
54.7 
54.7 
64.6 
64.6 
54.6 
54,5 
54.4 
54.4 
54.3 
54.3 
54.3 
54.2 
54.2 
54.1 
54.1 
64.0 
54.0 
53.9 
53.9 
53.8 
53.8 
53.7 
53.7 
53.6 
53.6 
53.6 
53.5 
53.5 
53.4 
53.4 
63.3 
63.3 
63.2 
53.2 
53.1 
63.1 
63.1 
53.0 
53.0 
52.9 
52.9 
52.8 
52.8 
52.8 
52.7 
52.7 
52.6 
52.6 
52.6 
52.5 
52.4 
62.4 
52.3 
52.3 
52.3 
52.2 
52.2 
52.1 


Cosine. 

9.970152 
970103 
970055 
970006 
969957 
969909 
969860 
9()98ll 
969762 
969714 
9()9665 
.9()9616 
969567 
969518 
969469 
969^^20 
969370 
969321 
969272 
969223 
969173 

9.969124 
969076 
969025 
968976 
968926 
968877 
968827 
968777 
968728 
968678 

9.968628 
968578 
968528 
968479 
968429 
968379 
968329 
968278 
968228 
968178 

9.968128 
968078 
968027 
967977 
967927 
967876 
967826 
967775 
967725 
967674 

9.967624 
967573  i 
967522 
967471 
967421  i 
967370 
967319 
967268 
967217 
96.166 
"sine; 


D.  10' 


8.1 
8.1 
8.1 
8.1 
8.1 
8.1 
8.1 


1 
1 
1 
2 
2 
2 
2 
2 
2 
8.2 
8.2 
8.2 
8.2 
8.2 
8.2 
8.2 
8.2 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.5 


Tang. 


9.584177 
,584565 
584932 
585309 
586686 
586062 
686439 
686815 
687190 
687666 
587941 

9.588316 
588691 
6890o6 
589440 
689814 
590188 
690562 
590935 
691308 
591681 
692054 
592426 
692798 
593170 
593542 
593914 
594286 
694666 
595027 
595398 

9.695768 
69ol38 
696508 
596878 
597247 
597616 
597985 
598364 
598722 
599091 

9.599459 
699827 
600194 
600562 
600929 
601296 
601662 
602029 
602395 
602761 

9.603127 
603493 
603858 
604223 
604588 
601953 
605317 
605682 
t  06046 
606410 
Cotang. 


D.  10"i     Cotanir.      iN.sine.  N.  cc8, 


62.9 
62.9 
62.8 
62.8 
62.7 
62.7 
62.7 
62.6 
62.6 
62.6 
162.5 
62.5 
62.4 
62.4 
62.3 
62.3 
62.3 
62.2 
62.2 
62.2 
62.1 
62.1 
62.0 
62.0 
61.9 
61.9 
61.8 
61.8 
61.8 
61.7 
61.7 
61.7 
61.6 
61.6 
61.6 
61.6 
61.5 
61.5 
61.4 
61.4 
61.3 
61 
61 
61 
61 
61 
61 
61.1 
61.0 
61.0 
61.0 
60.9 
60.9 
60.9 
60.8 
60.8 
60.7 
60.7 
60.7 
60.6 


10.415823 
415445 
415068 
414691 
414314 
413938 
413561 
413185 
412810 
412434 
412059 

10.411684 
411309 
410934 
410560 
410186 
409812 
409438 
409065 
408692 
408319 

10.407946 
407574 
407202 
406829 
406458 
406086 
405715 
405344 
404973 
404602 

10.404232 
403862 
403492 
403122 
402753 
402384 
402015 
401646 
401278 
400909 

10.400541 
400173 
399806 
399438 
399071 
398704  i 
398338 
397971 
397605 
397239 

10.396873 
396507 
396142 
395777 
395412 
395047 
394683 
394318 
393954 
393590 
Tang. 


35837 
i  35864 
35891 


93368 
93348 
93337 


35918193327 
35945  93316 
36973  [93306 
36000  ;93295 
'  36027  ;93285 
136054193274 
|3608Ii93264 


36108 
136135 
136162 
: 36190 


3621' 


93253 
93243 
93232 
93222 
93211 


136244  93201 
113627193190 
1 1 36298  J93 180 
;  36325^3169 
!!36352;93159 
ii36379|93l48 
1 1 36406 193 137 
136434193127 
I' 364611931 16 
i  36488 193106 
136515193095 
i  36542193084 
1136569193074 
!|36596  930t)3 
1136623  93052 
3665093042 
I !  3667-7  93031 
;  36704193020 
ij  36731 193010 
|l36758j92999 
36785 192988 
;  368 12 192978 
1 1 36839  92967 
36867  92956 
36894  92945 
3692192936 
136948  92926 
36975  92913 
!  37002  92902 
,37029  92892 
37056  92881 
37083  92870 
37110192859 
37137  92849 
'37164  92838 
3719192827 
37218  92816 
37245  92805 
37272  b2794 
37299  92784 
37326192773 
37353  92762 
3738092751 
37407  92740 


37434 
37461 


92729 
92718 


N.  COS.  N.pinc. 


60 
59 
58 
57 
66 
65 
64 
53 
52 
51 
I  50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
3  7 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
26 
24 
23 
22 
21 
20 
19 
18 
17 
16 
16 
14 
13 
12  , 
11 

10  ; 

9 

8 

7 


68  Degrees. 


TABLE  II. 


Log.  Sines  and  Tangents.    (22°)    Natural  Sinea. 


T\ 


Sine. 


9.573575 
67SyS8 
574200 
574512 
5748-24 
575 13() 
675447 


Drm 


52. 
52. 
52. 
51. 
51. 
51, 
51. 

57(iUfi9!?J- 
57(i379^,- 
57GG89J^, 
9.57(^999  I  °, 
577309 '  ?, 
577t)18i?|- 
577927  ?  ■ 
578236  ""^ 


578545 
578853 
579162 
579470 
579777 
9.580085 
580392 
680099 
581005 
581312 
681618 
681924 
582229 
582535 
582840 
583145 


583449 1 
583754  ""■ 
684058  J"" 
584361^"- 
584665  I  J,, 
584968  i  J,,' 

50. 
50. 
50. 
50. 
60. 
50, 
150. 
50. 


685574 
685877 
9.686179 
586482 
686783 
687085 
587386 
587688,,  , 
68  7989!?^; 
688289  I ?^ 
588590 ! ?" 
588890  ^" 


9.689190 
689489 
589789 
590088 
.•90387 
690686 
5909-^4 
691282 
69158(> 
5;,»1878 
Co.sine. 


CJosine. 


D.  lU" 


9.967166 
967115 
967064 
967013 
966961 
9W)910 
96^i859 
966808 
966756 
96<)705 
9(i6t)53 

9.9()()602 
9()655() 
966499 
966447 
9<>6395 
96(>344 
96()292 

-966240 

^9t;6188 
966136 

9.9(>6085 
9t)6>i33 
965981 
965928 
965876 
965824 
965772 
965720 
965t)68 
965()15 
9655(J3 
965511 
965458 
9()6406 
966353 
965301 
965248 
965195 
965143 
965090 
965037 
964984 
9t)4931 
964879 
964826 
964773 
964719 
964666 
964613 
964560 
9G4507 
964454 
964400 
964347 
964294 
964240 
964187 
964133 
W)4080 
964026 


Sin.i 


8.5 
8.6 
8.6 
8.5 
8.6 
8.6 
8.6 
8.5 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 


8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 


Tang. 

|.60<>410 
60<J773 
607137 
607500 
607863 
608225 
608688 
608960 
609312 
609674 
610036 

1.610397 
610759 
611120 
611480 
611841 
612201 
612561 
612921 
613281 
613641 

1.614000 
614359 
614718 
616077 
615435 
615793 
616151 
616509 
616867 
617224 

.617582 
617939 
618295 
618652 
619008 
619364 
619721 
620076 
620432 
620787 

1.621142 
621497 
621852 
622207 
622561 
622915 
623269 
623623 
623976 
624330 

1.624683 
625036 
625388 
625741 
K26093 
o26445 
626797 
627149 
627601 
627852 

Co  tang. 
iK-greeff. 


D.  10^^ 

60.6 
60.6 
60.5 
60.5 
60.4 
60.4 
60.4 
60.3 
60.3 
60.3 
60.2 
60.2 
60.2 
60.1 
60.1 
60.1 
60.0 
60.0 
60.0 
59.9 
59.9 
59.8 
59.8 
59.8 
69.7 
59.7 
59.7 
59.6 
59.6 
59.6 
59.5 
59.5 
59.5 
59.4 
59.4 
59.4 
59.3 
59.3 
59.3 
59.2 
59.2 
59.2 
59.1 
59.1 
59.0 
59.0 
69.0 
58.9 
58.9 
58.9 
58.8 
58.8 
58.8 
58.7 
58.7 
58.7 
58.6 
58.6 
58.6 
58,5 


37461 
37488 
37515 
37542 
37569 
37595 
37622 


Cotang.       NTsine.  N,  cos. 

10.393590 

393227 

392S63 

392500 

392137 

391775 

391412 

391050 

390688 

390326 

3899()4 
10.389603 

389241 

388880 

388520 

388159 

387799 

387439 

387079 

386719 

.386369 
10.3860(»0 

385641 

385282 

384923 

384565 

384207 

383849 

383491 

383133 

382776 
10-382418 

3820ol 

381i05 

381348 

380992 

380636 

380279 

379924 

379568 

379213 
1  J. 378858 

378503 

378148 

377793 

377439 

377085 

376731 

376377 

376024 

375670 
10-37.5317 

374964 

374612 

374259 

373907 

373555 

37G263 

373851 

372499 

372148 


92718 
92707 
92697 
92U86 
92ti75 
92664 
92653  I  54 
37649|92()42  53 
37676192631  62 
37703|92620  51 
37730|92609  50 
37757  92598  149 
37784'92.587  48 
37811'92576|47 
37838  92565  4(i 
37865  92554  45 
37892:92543  144 
37919  92532  143 
3794692521  |  42 
37973  92510  141 
37999  92499 
38026^92488 
38053  92477 
'  38080  92466 
1 3810792455 
38134  92444:35 
J381()l  92432  134 
38188  92421  |  33 
38215  92410  32 
I  38-241!  92399  1  31 
1 38268  92388  1 30 
38295  9-2377  129 
:  383-22  92366  28 
i  38349192355  27 
; 3837692343  j 26 
38403  92332  '  25 
138430  92321  [24 
138456  92310  123 
138483  9-2299  22 


38510  92287 


|38537!92276 
j  38564  92265 


38591  92254 
38617  92243 
'3864492231 
3867192220  115 
38<i98  92209 '14 


13 


Tang. 


38725  92198 
,  38752  92-i  86!  12 
38778  92175  I  11 
:  38805  92164 
38832  92152 
38859  92141 
38S86  92130 
38912  92119 
38939  92107 
38966  92096 
38993  92085 
39020  92ij,  3 
,  3904«)  9-2062 
J39073|92050 
I  X.  cos  J  N.8ine. 


44 


Log.  Sinea  and  Tangents.  (23°)  Natural  Sines. 


TABLE  II, 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 


Sine. 

9.591878 
59217<i 
592473 
592770 
6930o7 
593363 
593659 
593955 
594251 
594547 
594S42 

9.595137 
695432 
595727 
59J021 
596315 
596609 
696903 
597196 
597490 
597783 

9.598U75 
598368 
598660 
598952 
599244 
699536 
599827 
600118 
600409 
600/00 

9.600990 
601280 
601570 
601860 
602150 
602439- 
602728 
60301 7 
603305 
603594 

9.603882 
604170 
604457 
604/45 
605032 
605319 
605606 
605892 
60.il  79 
60-O465 

9.606751 
607036 
607322 
607607 
60/892 
608177 
608461 
60874) 
609029 
609313 
C!osine.  | 


D.  W 


49.6 

49.5 

49.5 

49.5 

49.4 

49.4 

49.3 

49.3 

49.3 

49.2 

49.2 

49.1 

49.1 

49.1 

49.0 

49.0 

48.9 

48.9 

48.9 

48.8 

48.8 

48.7 

48.7 

48.7 

48.6 

^18.6 

48.6 

48.5 

48.5 

48.4 

48.4 

48.4 

48.3 

48.3 

48.2 

48.2 

48.2 

48.1 

48.1 

48.1 

48.0 

48.0 

47.9 

47.9 

47.9 

47  8 

47.8 

47.8 

47.7 

47.7 

47.6 

47.6 

47.6 

47.5 

47-5 

47.4 

47.4 

47.4 

47  3 

47.3 


Cosine. 

1.964026 
963972 
903919 
9G3865 
963811 
963757 
963704 
963650 
963596 
963542 
963488 
1.963434 
963379 
963325 
963271 
963217 
•  963163 
963108 
9()3054 
962999 
962945 
.962890 
962836 
962781 
962727 
9(r2672 
962617 
962562 
962508 
962453 
962398 
.962343 
9()2288 
962233 
962178 
962123 
962067 
962012 
961957 
961902 
961846 
.961791 
961735 
961680 
961624 
961569 
961513 
961458 
961402 
961346 
961290 
,961235 
961179 
961123 
961037 
961011 
9(i0955 
9t>0899 
960843 
960786 
9 JO 730 


"I   Sine. 


8.9 

8.9 

8.9 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9  2 

9.2 

9.2 

9.2 

9.2 

9.2 

9.2 

9.2 

9.2 

9.2 

9.2 

9.2 

9.2 

9.2 

9.3 

9.3 

9.3 

9.3 

9,3 

9.3 

9.3 

9.3 

9.3 

9.3 

9.3 

9.3 

9.3 

9.3 

9.4 

9.4 

'i'ang. 

9.627852 
628203 
628554 
628905 
629256 
6296U6 
629956 
630306 
630656 
631005 
631355 
.631704 
632053 
632401 
632760 
633098 
633447 
633795 
634143 
634490 
634838 
9.635186 
635532 
635879 
636226 
636572 
636919 
637266 
637611 
637956 
638302 
). 638647 
638992 
639337 
639682 
640027 
640371 
640716 
641060 
641404 
641747 
1.642091 
642434 
642777 
643120 
643463 
643806 
644148 
644490 
644832 
645174 
,64.5516 
645857 
64(il99 
646540 
646881 
647222 
647562 
647903 
648243 
648583 
Cotang. 


D.  10"|  Coking,   N.  8ine.|N.  cos, 


58.5 

68.5 

58.5 

58,4 

58.4 

68.3 

58,3 

58.3 

58.3 

68.2 

68.2 

68.2 

68,1 

58.1 

68.1 

58,0 

68.0 

58.0 

57.9 

67.9 

67.9 

57.8 

67.8 

67.8 

67.7 

57.7 

57.7 

57.7 

67,.  6 

57.6 

57.6 

57,6 

57.5 

67.5 

67,4 

67,4 

57.4 

57.3 

67.3 

57.3 

57.2 

67.2 

57.2 

57.2 

57.1 

67.1 

67,1 

57.0 

67,0 

67.0 

56.9 

66.9 

56.9 

56.9 

56.8 

56.8 

66,8 

56.7 

56.7 

56.7 


10.372148 
371797 
371446 
371095 
370745 
370394 
370044 
369694 
369344 
368995 
368645 
10.368296 
367947 
367599 
367250 
36(J902 
366553 
366205 
365857 
36.5510; 
305162  I 
10.364815 
364468  - 
364121  I 
363774 ! 
363428 ! 
363081 j 
362735  I 
362389 1 
362044 ! 
3616981 
10.361353 j 
3610081 
360663  I 
360318! 
359973  i 
369629  I 
359284 1 
368940 
358596 
358253 
10.367909 
357566 
357223 
356880 
366537 
356194 
355852 
355510 
355168 
354826 
10.354484 
364143 
353801 
353460 
353119 
352778 
352438 
352097 
351757 
351417 
"fi^ng^ 


39G73 192050 


39100 
39127 
391.53 
39180 
39207 


92039 
92028 
92016 
92005 
91994 


39234  [91 982 
39260  91971 


39287 
39314 
39341 
39367 
39394 
39421 
39448 
39474 


91959 
91048 


91914 
91902 
91891 
91879 


39501  91868 
■91866 


3952H 
39555 
39581 
39608 
39636 
39661 
39688 


91845 
91833 
91822 
91810 
91799 
91787 


3971591775 
3974191764 


! 39768 
I  39795 
; 39822 
; 39848 
I  39875 
i  39902 
I  39928 
i  39955 
, 39982 
: 40008 
! 40035 


400(i2|9l625 
40088|91613 
40115191601 
40141  i91590 
40168J915.8 
40195  91666 


40221 
40248 
40275 


40301  91519 
40328:91508 
40355191496 
40381191484 
40408191472 
4043491461 
40461191449 
40488'91437 
40514  91425 


40641 
40567 
40594 


406211913/8 
40(J47  91366 
40674  913,55 
N.  COS.  N.sine, 


919S6  150 
91925  I  49 
48 
47 
46 
46 
44 
43 
42 
41 
40 
39 
38 
37 
36 
36 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 

8  I 

7 

6 

5 

4 

3 

2 

1 

0 


91752 
91741 
91729 
91718 
91706 
91694 
91683 
91671 
91660 
91648 
91636 


91555 
91543 
91531 


91414 
91402 
91o90 


66  Degrees. 


Log.  Sines  and  Tangents.  (24°)  Natural  Sines. 


45 


0 

1 
2 
3 

t 

6 

7 

8 

9 

TO 

II 

1-2 

13 

14 

15 

U> 

17 

18 

19 

20 

21 

22 

23 

24 

25 

2« 

27 

28 

29 

30 

31 

32 

33 

34 

35 

3(i 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

64 

55 

5(i 

67 

58 

59 

60 


Sino. 

GiD313 
(iO'Jo'Ji 
609 SSO 
t)10U)4 
(ilUU,- 
t)10.'29 
yilJl-2 
611294 
611576 
61 1858 
fi]-2l\W 
6124-21 
612702 
61298^ 
613264 
613545 
613825 
614105 
614385 
6146(i6 
614944 
9.615223 
615502 
616781 
616060 
616338 
616616 
616894 
617172 
617450 
617727 
6I80OI 
618281 
618558 
6IS834 
619110 
6193»6 
619662 
619938 
620213 
620488 

19.620763 
621038 
621313 
621587 
621861 
622135 
622409 
622082 
62-29 ')6 
623229 

19.6-23512 
6-23774 
624047 
624319 
624591 
624863 
625135 
62540J 
6'2.5u7  7 
625948 


D.  10' 


47.3 

47.2 

47.2 

47.2 

47.1 

47.1 

47.0 

47.0 

47.0 

4o.9 

46.9 

46.9 

46.8 

46.8 

46.7 

46.7 

46.7 

46.6 

46.6 

4«}.6 

46.5 

46.5 

46.5 

46.4 

46.4 

46.4 

46.3 

46.3 

46.2 

46.2 

46.2 

4<).l 

4(i.l 

46,1 

46.0 

46.0 

46.0 

45.9 

45.9 

45.9 

45.8 

45.8 

45.7 

45.7 

45.7 

45,6 

45.6 

45.6 

45,6 

45.5 

45.6 

45.4 

45.4 

45.4 

45.^ 

45.3 

46.3 

45.2 

45,2 

45.2 


Cosini;. 

1.960730 
960674 
9;K)6  18 
9o0561 
960505 
9;)0448 
960392 
96J335 
960279 
960222 
960166 
1.960109 
960052 
959995 
959938 
959882 
9598-25 
959768 
959711 
959654 
959596 
.959539 
959482 
959425 
95'J3(>8 
959310 
959-253 
959195 
959138 
959081 
9690-23 
.958965 
958908 
958850 
958792 
958 i 34 
958677 
958619 
958561 
958503 
958445 
.958387 
958329 
968271 
958213 
958154 
968096 
958038 
957979 
957921 
957863 
.957804 
95774(> 
95 7687 
957628 
95*570 
957511 
957452 
957393 
.■■5/335 
957276 


D.  10' 


Cosin 


9.4 
9.4 
9.4 
9.4 
9.4 
9.4 
9.4 
9.4 
9.4 
9.4 
9.4 
9.5 
y.6 
9.6 
9.5 
9.5 
9.6 
9.5 
9.5 
9.5 
9.5 
9.5 
9.6 
9.5 
9.5 
9.6 
9.6 
9.6 
9.6 


Tiin;^. 


6 

6 

6 

6 

6 

6 

6 

9.6 

9,6 

9.6 

9.7 


9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.8 
9.8 
9.8 
9.8 
9.8 
9.8 
9.8 
9.8 


9.648583 
648923 
649263 
64^)602 
64994-2 
650281 
650i)20 
650959 
651-297 
651636 
651974 
65-2312 
652()50 
65-2988 
65332(> 
6536(>3 
654000 
654337 
654174 
655011 
655348 
9.655684 
656020 
66(i356 
656692 
657028 
667364 
667699 
658034 
658369 
I  658(04 
[9.659039 
659373 
659708 
660042 
66(X3  76 
660710 
661043 
6<il377 
661710 
662043 
9.662376 
662709 
663042 
663375 
663707 
664039 
664371 
664703 
GtioOlid 
665366 
9.665(j97 
666029 
666360 
666691 
667021 
6(i7352 
667682 
668013 
668343 
t)686i2 


D.  10 


Coti.nt,'. 


56.6 

56.6 

56,6 

56  6 

56.5 

56.5 

59.6 

56.4 

56.4 

.56.4 

56.3 

56.3 

56.3 

56.3 

56,2 

56  2 

56,2 

56,1 

56,1 

56.1 

56.1 

56.0 

56.0 

56.0 

56.9 

55.9 

55.9 

56.9 

55.8 

55.8 

65.8 

55.8 

55.7 

55.7 

55.7 

55.7 

55.6 

65.6 

66.6 

55.5 

.55,5 

55,5 

55,4 

56.4 

55.4 

65,4 

55,3 

65.3 

55.3 

55.3 

65.2 

55.2 

55.2 

65.1 

55.1 

5.5. 1 

55. 1 

55.0 

55.0 

55.0 


Cotaiij;. 


10.351417 
351077' 
350737: 
360398 
350J58 
349; 19 
3493^0 
349041  I 
348703 
3483«)4 ! 
3480-26 

10.347688 
347350 
347012; 
346674 i 
346337  > 
346000  ; 
345663 
345326 
344989 
344652  j! 

10.344316 r 

343980 

343644 

343308' 

342972 

342636' 

342301 

341966 ^ 

341631  I 

341-296;. 

10.340961  '; 
340627 
3402921' 
3399581 
3396241 
339290  ii 
338957  I i 
338623  i 
3382901 1 
337957'; 

10.337624 
3372911: 
336958  jl 
3366251 
336293,' 
335961 
3366291 
335297 j 
334966 : 
334634  i 

10.3343031 
333971 ! 
3S3()->0 
33o309 ! 
3329,9! 
33'2648 1 
332318 
331987 
331667 
331328 


4080*  lb  1295 
408o3;91283 
408()U9l2/2 
40b86;9l260 
40913  91248 
40939  91:^36 
4n966!9r224 
4099^2  a  1-21-.; 
41019191200 
41045 '91 18» 
410/2191176 
41098;^  1 164 
411i>5  9ll52 
4115191140 
41178  911-28 
4120491 116 
4123191104 
41257  9109-,^ 
41284  91080 
413109106b 
41337  91056 
4136391044 
41390  91032 
41416  9lO*-iO 
4144391008 
41469  9099() 
41496  90984 
41522  9097-2 
41649  90960 
41576  90948 
41602  9093b 
41628  90924 
141655  90911 
i  41681  9089y 
I41707;9u88i 
,41734:9()8;5 
'417(i(l90863 
: 4178  7 90b51 
41813  90t5oU 
41840  908-^6 
41866  9O0U 
41892  90bO-z 
419199U-90 
41945  9077  b 
4 1.972  90 1 66 
41998  907  5o 
42U24  90i-il 
42051  90729 
42077  90/1/ 
42104  90704 
421o0  9069-^ 
421.56  9U-bO 
42l8o9lX>t)b 
42209  90655 
42-235  9vX>4o 
422(i2;90.,31 
N .  c<is.|N.siiic. 


65  iHHrreee. 


46 


Log.  Sines  and  Tangents.    (25"^)    Natural  Bines. 


TABLE  II. 


"Tang.      ID.  10"i     Cotang.     |  N.f"iiio.|N.  cos.i 

422(i2|90G31  Uo 


Sine.   D.  10"  Cosine.  •  D.  10" 


2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
19 
50 
61 
62 
53 
54 
55 
56 
57 
58 
59 
60 


.625948 
626219 
626490 
6267,0 
627030 
627300 
627570 
627840 
628109 
628378 
628647 


45 

45 

45 

45 

45 

45 

44 

44 

44 

144 

144 
628916  I  T^ 

629185  Jl 

629453  l^ 


629721 
629989 
630257 
630524 
630792 
631059 
631326 

.631593 
631859 
632125 
632392 
632658 
632923 
633189 
633454 
633719 
633984 

.634249 
634514 
634778 
635042 
635303 
635570 
635834 
636097 
636360 
636623 

.63i)8S6 
637148 
637411 
637673 
637935 
638197 
638458  I 
638/20 
638981 
639242 

|.6;J9503 
639764 
640024 
640284 
640544 
6 10804 
641ij64 
641324 
641584 
641842 

Cosine. 


9.957276 
957217 
957158 
957099 
957040 
956981 
956921 
956862 
956803 
956744 
956684 

9.956625 
956566 
9565 J6 
956447 
956387 
956327 
956268 
956208 
956148 
956089 

9.956029 
955969 
955909 
955849 
955789 
955729 
955669 
955609 
955548 
955488 
955428 
955368 
955307 
955247 
9r,5l86 
955126 
955065 
9550J5 
954944 
954883 
.954823 
954762 
954701 
954640 
954579 
954518 
954457 
954396 
954335 
954274 

9  954213 
954152 
954090 
954029 
953968 
953906 
953845 
953783 
953722 
953660 


9.8 
9.8 
9.8 
9.8 
9,8 
9.8 
9.9 
9.9 
9.9 
9  9 


9.9 
9.9 
9.9 
9  9 

9.y 

9.9 
9.9 
9.9 
10. 0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10. 0 
10 
10 
10 
10 
10 
10 

10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.3 


9.668673 
669002 
609332 
669{)61 
669991 
670320 
670649 
670977 
671306 
671634 
671963 

9.672291 
672619 
672947 
673274 
673602 
673929 
674257 
674584 
674910 
675237 

9.675564 
675890 
676216 
676543 
676869 
677194 
677520 
677846 
678171 
678496 

9.678821 
679146 
679471 
679795 
680120 
680444 
680768 
681092 
681416 
681740 

9.682063 
68-'387 
682710 
683033 
683366 
683679 
684001 
684324 
684646 
684968 
1.685290 
685612 
685934 
686255 
686577 
686898 
687219 
687540 
6878(>1 
688182 
Cotang. 


65.0 
54.9 
64.9 
54.9 
54.8 
54.8 
64.8 
54.8 
54.7 
54.7 
64.7 
54.7 
64.6 
54.6 
64.6 
54.6 
54.5 
64.6 
54,5 
54.4 
64.4 
64.4 
54.4 
64.3 
54.3 
64.3 
54.3 
54.2 
54.2 
54.2 
64.2 
64.1 
54.1 
64.1 
54.1 
54.0 
54.0 
54.0 
64.0 
53.9 
63.9 
53.9 
53.9 
53.8 
53.8 
63.8 
53.8 
53.7 
53.7 
53.7 


&). 331327 


330998 ■ 
330368 
330339 
330009 ; 
329680 
329351 ' 
329023 
328694 ' 
328366  i 
328037  I 

10.3277091 
327381 i 
327053  ! 
326726 ; 
326398 ■ 
326071 
325743 : 
325416 1 
325090 
324763' 

10.324436! 
324110 
323784' 
323457: 
323131 
322806: 
322480! 
322154 
321829  i 
321 504 : 

10.321179 
320854' 
320529 
320206 
319880 
319556 
319232, 
318908 
318584 
318260 

10.317937 
317613 
317290 
316967 
316644 
316321 
315999 
315676 
315354 
-o  7  315032 
°^i'r  10.314710 
314388 


63.6 

53.6 

53.6 

53 

63 

53 

53 

53 

63 


4228b 
42315 
42341 
42367 
42394 
42420 
42446 
42473 
42499 
42525 
42652 
42578 
42604 
42631 
42667 
42683 
42709 
42736 
42762 
42788 
42816 
42841 
4286/ 
42894 


yj613 !69 
90()06  I  68 
90594  1 57 
90582  i  66 
90669 { 65 
90557  1 54 
90545  j  63 
90532  '  52 
90520; 61 
905(i7  I  50 
90495  1 49 
90483  ,  48 
904/0147 
90458  i  46 
90446  1 45 
90433  1 44 
90421  !43 
90408  1 42 


90396 
90383 
90371 
90368  }  38 
90346  37 


90334 


42920J90321 
42946  y030ii 
42972  90296 
42999  90284 
43025  90271 
43051  90259 
43077  90246 
43104i90233 
43 130j  90221 
43156190208 
431S2'90196 
43209190183 
43235  90171 
43261  90158 
43287190 146 
43313190133 
43340190120 
43366190108 
43392190095 
43418|900ei2 
434451900/0 


43471 
43497 
43623 
43549 


90057 
9004c 
90032 
90019 


43575  90007 


314066; 

313745 

313423 

313102 

312781 

3124o0 

312139 

3118181 


43602 
.43, 2b 
43654 
43680 
4370O 
43733 
43/59 
43785 


89994 
89981 
89968 
89956 
8994c 
89930 
89918 
89905 


Tang. 


^3bli;8yb92 
43837  89879 
N.  cos.  .\.8ir<'. 


fi4  Deirrctis. 


Lo}?.  Sines  and  Tangents.    (26°)    Natural  Sines. 


47 


Sine. 


0 
1 
o 

3 
4 
5 
G 
7 
8 
9 
lU 
U 
1-2 
13 
14 
15 
Hi 
17 
IH 
1!) 
20 
21 
22 
23 
24 
26 
26 
27 
28 
29 
30 
31 
3-2 
33 
34 
35 
36 

3; 

3rt 

3y 

40 
41 
42 
4^J 
44 
45 
46 
47 
48 

4y 

60 
61 
62 
63 
64 
55 
66 
6< 
58 

6y 

60 


D.  lO'l    Cosine.     D.  lo" 


9.641842 
642101 
6423ti0 
642618 
642877 
643135 
643393 
643()50 
643908 
644165 
(>44423 

*J  644tJ80 
644936 
645193 
645450 
645706 
615962 
646218 
646474 
646729 
64<i984 

9.647240 
647494 ! 
647749 
648004 
64.S258 
648512 
648766 
649020 
6492? 4 
64:i527 

9. 649 /hi 


650J34j^:j 
650287  If 
650539 ;  7Z 
650,92,*'; 

651044 ;^r 

651297 ;4j 

6515  49:^;; 
65ihoo,;r. 

652052  ^^ 


9.65-.^304 
652555 
652806 
65305  7 
653308 
65o558  ; 
65381.8 
6540J9 
654;iOJ 
(i54558  , 

£  ,6.34808  , 
655058  j 
65530/ j 
655556  I 
(»5;J8U5  I 
65o054 
656302 \ 
65*i5Jl  i 
(i'iyt'i'J'J  ; 
65.017 j 
Cos  i  lie.  I 


9.953660 
953699 
953537 
953475 
953413 
953362 
953290 
953228 
953166 
963104 
953042 

9.952980 
952918 
952856 
952793 
952731 
952669 
952606 
952544 
952481 
952419 

9.952356 
952294 
952231 
952168 
952106 
952043 
951980 
951917 
951&54 
951791 
951728 
951665 
951602 
951639 
951476 
951412 
951349 
961286 
9512-.^2 
951159 
961090 
951032 
950968 
950905 
95J841 
9507/8 
950714 
950650 
950586 
950522 
.950458 
950394 
950350 
950->66 
95J202 
950 13h 
9500 i 4 
950011) 
949945 
949861 


9.96 


10.3 

10.3 

10.3 

10.3 

10.3 

10.3 

10.3 

10.3 

10.3 

10.3 

10.3 

10.4 

10.4 

10.4 

10.4 

10.4 

10.4 

10.4 

10.4 

10.4 

10.4 

10.4 

10.4 

10.4 

10.5 

10.5 

10.6 

10.5 

10.6 

10.5 

10 

10 

10 

10 

10 

10 

10.5 

10.6 

10,6 

10.6 

10.6 

10.6 

10,6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.7 

10.7 

10.7 

10.7 

10.7 

10.7 

10.7 

10.7 

10.7 

l\i.7 


Tang. 


D.  10' 


63.4 
53.4 
63.4 
53.3 
53.3 
63.3 
53.3 
53.3 
63.2 
63.2 
53.2 
63.1 
53.1 
53.1 


.688182 

(588502 

688823 

689143 

6894^>3 

689783 

690103 

690423 

690742 

691062 

691381 
.691700 

692019  I 

692338  I 

692o66  L.>  , 

6929751^:^- 

693293  °^-\ 

693612  ^3-^ 

693930  I  °^H 

694248 

694566 
,694883 

695201 

695518 

695836 

696153 

696470 

696787 

697103 

697420 

697736 
.698053 


63.0 
53.0 
52.9 
62.9 
52.9 
62.9 
62.9 
52.8 
[52.8 
[52.8 
!52.8 
152.7 
152.7 

698369   ^'i'l 

52.6 
52.6 
52.6 
52.6 
62.6 
52.6 


698(i86 
699001 
699316 
699632 
(i99947 
700-63 
70vi6i8 
700893 

•.701-.^08 
701523 
701837 
702162 
70.;4b6 
702,80 
703095 
703409 
703,23 
704036 

1. 701350 
704O63 
704977 
705290 
705(i03 
705916 
70i/i28 
70t641 
70v*h54 
70i lo6 

fcliiiig- 
(41  iH'grees. 


52.5 
52.6 
62.4 
52.4 
62.4 
52.4 
52.4 
52.3 
52.3 
52.3 
52.3 
52.2 
52.2 
52.2 
52.2 
52.2 
52.1 
5-J.l 
52.1 
52 . 1 
52.1 


CoUmg.       N.  sine.  N.  eo«. 


10.311818 
311498 
311177 
310857 
310537 
310217 
309897 
309577 
309258 
308938 
308619 

10.308300 
307981 
307662 
307344 
307026 
306707 
306388 
306070 
305752 
305434 

10.305117 
304799 
304482 
3041()4 
303847 
30J5oO 
303213 
302897 ' 
302680 
302264 : 

10-301947 
301631 ' 
301315 
300999; 
300()84 
300368 
300053 
299737 : 
-299422 ' 
299107  i 

10-2 :8<92! 
'z98477  j 
298163 
2:r,848  ■ 
29 < 634' 
29,220 
29()9U6 
2i'6691  ; 
ic9(i277  I 
2959(54  I 
295650 
295337  I 
295023  I 
294710; 
294:^97  j 
2940h4 ! 
t:93772  i 
293469 1 
29M46 
292834 
Tang. 


89879100 
89867  69 
89854  58 
89841 157 
898-28  I  56 
89816  155 
43994  89803  64 
44020  89790  53 
89777  !  52 
44072  897(i4j  61 
44098  89752  60 
89739  49 
89726  48 
89713 
89.00 
89(i87 
89674 
89()62 
89649  j  42 
144333  89636  1 41 
i  44359  89(>23  '  40 
8!)610i39 
89697  38 
89584  I  37 
89571  136 
89558  1 35 
89545  1 34 
89532 \ 33 
89519  132 
89506  31 
89493  1 30 
89480  ''  29 


44124 
44151 
44177 
44203 
44229 
44255 
44281 
44307 


I  44385 
44411 
: 44437 
{44464 
' 44490 
j 445 16 
j  44542 
I  446(.8 
I  44594 
1 44620 
I  44646 
144672 
! 44698 
I  44724 
44750 
i  44776 
! 44802 


89467  128 
89454  27 


10 


89441 [26 
89428  : 26 
8b4l6|24 
89402  23 

44828  89389,22 
l44854jH9376t2\ 
i  44880189303  ;  20 
'  44  91 R.  189360.  19 
|4493-j!89337!18 
i4495hih;M24  ;  17 
1 449841893 II  ;  16 
■45010:89298    15 

460y(ii89285    14 

450,,'jjh92,2;i3 

45088ih9259  12 
i45114i89245|ll 
i  45 1 40l89232  1 0 
'4516(iih9219|  " 
|45l92ih9206 
l46218!^9193 

46343  h9 180 
!4.o2()9,IS9l')7 

45296  h9 153 
'453vl:hyl40 

45347  h9 127 


453  73 

S.  COS.  N.sine. 


89114 
89101 


48 


liOfT.  SiiKis  and  Tangents.     (27°)     Natural  Sines. 


TABLE  II. 


a 


0 

1 

2 
3 
4 
6 

6 

7 

8 

9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21  9 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

40  I 

41  ;9 
42 
43 
44 
45 
16 
47 
48 
49 
50 
61 
62 
53 
54 
55 
5() 
67 
58 
b'J 
60 


657017 
()67295 
657542 
657790 
658037 
658284 
658531 
658778 
659 J25 
659271 
659517 
659763 
660J09 
6(i0255 
660501 
660/46 
660991 
66123a 
661481 
661726 
6619/0 
.662214 
662459 
662703 
66294tj 
663190 
663433 
663677 
663920 
664163 
6t)4406 
.6641)48 
664891 
665133 
665375 
665617 
665859 
666100 
666342 
()66583 
6(i6824 
.667065 
667305 
667516 
667786 
668027 
668267 
668o0<) 
668746 
668986 
669225 
669464 
6t>9703 
669942 
6701K1 
670419 
670)58 
670896 
671134 
671372 
6/1609 
Cfxsino. 


I),  lu' 

41.3 

41.3 

41.2 

41.2 

41.2 

41.2 

41.1 

41.1 

41.1 

41.0 

41.0 

41.0 

40.9 

40.9 

40.9 

40.9 

40.8 

40.8 

40.8 

40.7 

40.7 

40.7 

40.7 

40.6 

40.6 

40.6 

40.5 

40.6 

40.5 

40.5 

40.4 

40.4 

40.4 

40.3 

40.3 

40.3 

40.2 

40.2 

40.2 

40.2 

40.1 

40.1 

40.1 

40.1 

40.0 

40  0 

40.0 

39.9 

39.9 

39.9 

39.9 

39.8 

39.8 

39.8 

39.7 

39-7 

39.7 

39.7 

39  6 

39.6 


L 


L/U.smo. 

.9498.S1 
94Jrilj 
949752 
94  ^688 
949623 
949558 
949494 
949429 
949364 
949300 
949235 

1.949170 
949105 
949040 
948975 
948910 
948845 
948780 
948715 
948650 
948584 

1.948519 
948454 
948388 
948323 
948257 
948192 
948126 
948060 
947995 
947929 

1.947863 
947797 
947731 
947666 
947600 
947533 
947467 
947401 
947335 
9472(>9 

1.947203 
947 '36- 
947070 
947004 
946937 
946871 
946804 
946738 
94(>671 
946604 

^ 946538 
946471 
946404 
946337 
946270 
946203 
946136 
946069 
94tJ002 
945935 
Sine. 


8 
8 
,8 
8 
8 
8 
8 
8 
8 
8 
8 
,8 
8 
.9 
.9 
,9 
,9 
.9 
.9 
.9 
,9 
,9 
,9 
,9 
,9 
,0 
,0 
.0 
.0 
,0 
,0 
.0 
,0 
.0 
.0 
.0 
.0 
.0 

.1 
.1 
.1 
.1 
.1 
.1 
.1 
.1 
.1 
.1 
.1 
.1 
.1 

.2 

.2 

11.2 

11.2 


1).   lu  '   Colau-;.   N.  siiK; 


9.707166 

707478 

707790 

708102 

708414 

708726 

709037 

70J349 

709660 

709971 

710282 
9.710593 

71 0904 

711215 

711525 

711836 

712146 

712456 

712766 

713076 

713386 
9.713696 

714005 

714314 

714624 

714933 

715242 

715651 

715860 

716168 

716477 
9.716785 

717093 

717401 

717709 

718017 

7183-5 

718633 

718940 

719248 

719555 
9.719862  51.2 

720169 

720476 

720783 

721089 

721396 

721/02 

722009 

722315 

722621 
9,722927 

723232 

723538 

723^44 

724149 

724454 

724759 

7250-)5 

725369 

725674 
"Cotanj;. 


52.0 
52.0 
52.0 
52.0 
51.9 
51.9 
51.9 
51.9 
51.9 
61.8 
61.8 
51.8 
61.8 
51.8 
51.7 
51.7 
61.7 
61.7 
51.6 
61.6 
61.6 
51.6 
51.6 
61.5 
51.6 
61.5 
61.5 
51.4 
61.4 
61.4 
61.4 
51.4 
61.3 
61.3 
61.3 
61.3 
61.3 
51.2 
61.2 
61.2 


61.2 
51.1 
61.1 
51.1 
51.1 
61.1 
61.0 
51.0 
51.0 
51.0 
61.0 
60.9 
50.9 
50.9 
50.9 
60.9 
50.8 
60.8 
60.8 


N.  cos, 


89101  I  60  I 
89J87  ■  69 


10.292834  45399 
292522  45425 
292210  45451  89074 
291898  45477  89061 
291586  45503  8904« 
291274  45529 !890o5 
2909;)3  45554  !890il 
2901)51  45580  89U08 
290340  45t)06  88995 
290029  45632 188961 
289718  45658188968 

10.289407  i  46684J88955 
289096;  4571 0|88942 
288785  45736188928 
288475'  45  762  J889 15 
288164  45787  88902 
28/854  458 13 188888 
287644  45»39i888  75 


286924  45891188848 
286614'  45917188835 

10 .  286304  45942188822 
285995  45968188808 
286686  :  45994188795 
285376  ;  46020188782 
285067':  4604688768 
284758'  46072188  765 
284449!;  46097 188741 
284140;  46123188728 
283832  i!46149j887 16 
283523-46175188701 

10.283215  4620188688 
282907  j' 4622b  88674 
282599  j!  46252  88661 
282291  1146278  88647 
2819831' 46304  88634 
281676!  4()330  88620 


281367  II  46365 


88607 


28 1 060!!  46381 

280752/46407 

280445  46433 

10.280138  '4t)4o8 

279»31  4u484  88539 
279524  46510i88526 
279217|i4lj536bh51 


88593 

88680 
8S566 
88653 


278911!!  46561 


278604 
278298 
27/991 
277686 
277379 
10.277073 
276768 


46587 

!4(;613 

I  46639 
j  46664 
I  46690 
j  467 16 
46742 


88499 
88485 
884/2 
88458 
88445 
«843l 
88417 
88404 


2764u2i:  46767  88390. 


276156 
2  75851 
275546 
275241 


46793  8837/ 
46819  88363 
4684,^88349 
4b87W  88336 
274^)36  1  41,^96188322 
274031  4jk>92ll88308 
2  74326    4t)94/|88'.^95 


Tanji 


S.  roK.  -^-'-'iJ' 


<)2  Df^gret'S. 


TABLE  II. 


Log.  Sines  and  Tangents.    (28°)     Natural  Sines. 


49 


0 
I 

2 
3 
4 
6 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
l(i 
17 
18 
19 
20 
21 
22 
23 
24 
25 
21) 
27 
28 
29 
30 
31 
32 
33 
34 
35 
3G 
37 
38 
39 
40 
41 
42 
43 
44 
45 
Mi 
47 
48 
49 
60 
31 
i  52 
53 
54 
55 
5(> 
67 
68 
69 
60 


Siuu. 

.6716)9 
671847 
6720S4 
672321 
672558 
672795 
673032 
6732,iH 
673505 
673741 
673977 

.674213 
674448 
674684 
674919 
675155 
()75390 
675  i24 
675859 
6761)9  I 
676328 

.676562 
67679o 
677030 
6772()4 
677498 
677731 
677964 
678197 
678430 
6786i)3 

.678895 
679128 
679360 
679592 
679W24 
680.)5t) 
680288 
680519 
680750 
68iJ982 

.681213 
()81443 
68lt)74 
681905 
682135 
6823()5 
()82595- 
68-^825 
683055 
683284 

.683514 
683743 
6839-2 
684201 
684430 
684()58 
684887 
6851 15 
685343 
685571 

Oo.sine. 


D.  10' 


39.6 
39.6 
39.5 
39.6 
39.5 
39.4 
39.4 
39.4 
39.4 
39.3 
39.3 
39.3 
39.2 
39.2 
39.2 
39.2 
39.1 
39.1 
39.1 
39.1 
39.0 
39.0 
39.0 
39.0 
38.9 
38.9 
38.9 
38.8 
38.8 
38.8 
38.8 
38.7 
38.7 
38.7 
38.7 
38.6 
38.6 
38.6 
38.5 
38.5 
38.5 
38.5 
38.4 


Cosini' 


38.4 
38.4 
38.4 
38.3 
38.3 
38.3 
38,3 
38.2 
38.2 
38.2 
38.2 
38.1 
38.1 
38.1 
38.0 
;i8.0 
38.0 


1.945935 
9458(iS 
94580J 
945733 
94566;; 
945598 
9)5531 
945464 
945396 
945328 
945261 

1.945193 
945125 
945058 
944990 
944922 
944854 
944786 
944718 
944()50 
944582 

.944514 
94444<j 
944377 
944309 
944241 
944172 
944104 
944036 
943967 
943899 

^ 943830 
943761 
943(i93 
943624 
943555 
943486 
943417 
943348 
913279 
943210 

1.943141 
943072 
943003 
942934 
942864 
942795 
94272*) 
94265() 
942587 
942617 

.942448 
942378 
94230.8 
94-J239 
942169 
942099 
942029 
941959 
941889 
941819 
Sine. 


D.  10"   Tang.   D.  10"  Cotarig.   N.  sino.jN.  co. 


9.725674 
725979 
726284 
72()588 
72()892 
727197 
727501 
727805 
728109 
728412 
728716 

9.72902C 
729323 
729626 
729929 
730233 
730535 
73.J838 
731141 
731444 
731746 

9.732048 
732351 
732653 
732955 
733257 
733558 
733860 
734162 
734463 
734764 

9.735066 
735367 
735668 
735969 
736269 
736570 
736871 
737171 
737471 
737771 
°;9. 738071 
°  738371 
738671 
738971 
739271 
739570 
739870 
740169 
740468 
740767 
1.741066 
741365 
741664 
741962 
742261 
742559 
742858 
74315!) 
743454 
743752 
CotiiiTiir 


50.8 
50.8 
50  7 
50.7 
50.7 
50.7 
50.7 
50.6 
50.6 
50.6 
50.6 
50.6 
50.5 
50-5 
50  5 
50. 6 
50.5 
50.4 
50.4 
50.4 
50.4 
50.4 
503 
50.3 
50.3 
50.3 
50. 3 
50.2 
50.2 
50-2 
50.2 
50.2 
50.2 
50.1 
50.1 
50.1 
50.1 
60.1 
50.0 
50.0 
50.0 
50.0 
50.0 
49.9 
49.9 
49.9 
49.9 
49.9 
49.9 
49.8 
49.8 
49.8 
49.8 
49.8 
49.7 
49.7 
49.7 
49.7 
49.7 
49.7 


10.274.326 
274021 
273716 
273412 
273108 
272803 
272499 
272195 
271891 
2715881 
271284 

10.270980 
270677 
270374 
270071 
2697(>7 
2694()5 
269162 
268859 
268556 
268264 

10.267962 
267649 
267347 
267045 
26(i743 
26(5442 
266140 
26.5838 
265537 
265236 

10.264934 
264633 
264332 : 
264031  ' 
263731 
263  130 
263129 
262829 
262529 
2(J2229 : 

10.261929 
261629 
261329 
261029 
260729 
260430 
260130 
259831 
259532 
259233 

10.258934 
258635 ' 
258336 
258038 
257739 
257441 
257142 
25(i844 
25664() 
256248 


T:in), 


4(5947  88295 
46973  88281 
4(5999188267 
^7024:882.54 
4.050  88240 
4707(i!88226 
47 1 01  [882 13 
47127188199 
471.53188185 
47178|88172 
4720468158 
47229 188144 
472.55188130 
47281  [881 17 
47306188103 
47332:88089 
47358^8075 
47383(88062 
47409 '88048 
47434,88034 
47460 188020 
47486 188006 
47511  87993 
47537  87979 
475(52  8  79(55 
47.588:87951 
4761487937 
47639187923 
47(i65 '87909 
47690  87896 
477J6'87882 
47741 '878(58 
477(57  87854 
47793  87840 
47818  87826 
47844  87812 
47869:87798 
4789587784 
47920187770 
47946i87756 
4797187743 
47997187729 
48022  87715 
48048  87701 
48073  8768  7 
4809987(573 
48124  87(559 
481 5U  87(545 
48175  87(531 
48201  87617 
48226187603 
48262187589 
48277187575 
483031875(51 
4832818754(5 
48364 187532 
48379J87518 
48405187504 
48430l87490 
48456187476 
48  181(874(52 
N.  COS. In  .sine. 


61  Degrees. 


50 


Log.  Sines  and  Tangents.    (29°)    Natural  Sines. 


TABLE  II. 


D.  10" 


0 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11, 
12 
131 
14 
1.5 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
56 
56 
57 
58 

5a 

60 


.685571 
685799 
686027 
686254 
686482 
686709 
686936 
687163 
687389 
687616 
687843 
688069 
688295 
688521 
688747 
688972 
689198 
689423 
689648 
689873 
690098 

.690323 
690o48 
690772 
690996 
691220 
691444 
691668 
691892 
692116 
692339 

.692562 
092785 
693008 
693231 
693453 
693676 
693898 
694120 
694342 
694564 

.694786 
695007 
695229 
695460 
695671 

.  695892 
696113 
696334 
696554 
696775 

i,69b9i^5 
697215 
697435 
697654 
697874 
698094 
698313 
6«8532 
698761 
698970 


38.0 
37.9 
37.9 
37.9 
37.9 
37.8 
37,8 
37.8 
37.8 
37.7 
37.7 
37.7 
37.7 
37.6 
37.6 
37.6 
37.6 
37.5 
37.5 
37.6 
37.5 
37.4 
37.4 
37.4 
37.4 
37.3 
37.3 
37.3 
37.3 
37.5 
37.2 
37.2 
37.1 
37.1 
37.1 
37.1 
37.0 
37.0 
37.0 
37.0 
36.9 
36  9 
3h:.9 
36.9 
36.8 
30.8 
36.8 
36.8 
36.7 
36.7 
36.7 
36.7 
36.6 
36.6 
36.6 
36.6 
36.5 
36.5 
36.5 
36.5 


Cosine.  }D.  lO^^I   Tang. 


Cosine. 


941819 
941749 
941679 
941609 
941539 
941469 
941398 
941328 
941258 
941187 
941117 
,941046 
940975 
940905 
940834 
940763 
940693 
940622 
940561 
940480 
940409 
,940338 
940267 
940196 
940126 
940054 
939982 
939911 
939840 
939768 
939697 
.939626 
939554 
939482 
939410 
939339 
939267 
939196 
939123 
939062 
938980 
.938908 
938836 
938763 
938691 
938619 
938547 
938476 
938402 
938330 
938268 
.938186 
938113 
938040 
937967 
937895 
937822 
937749 
937676 
937604 
937531 


11.7 
11.7 
11.7 
11.7 
11.7 
11.7 
11.7 
11.7 
11.7 
11.7 
11.7 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 


Sine. 


11.8 
11.8 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
12.0 
12,0 
12.0 
12,0 
12,0 
12.0 
12.0 
12.0 
12.0 
12.0 
12,0 
12.0 
12,1 
12.1 
12,1 
12,1 
12.1 
12,1 
12,1 
12,1 
12.1 
12.1 
12.1 
12.1 


9.743752 
744050 
744348 
744645 
744943 
745240 
745538 
745836 
746132 
746429 
746726 
747023 
747319 
747616 
747913 
748209 
748505 
748801 
749097 
749393 
749689 

9.749985 
750281 
750576 
760872 
751167 
751462 
751757 
762052 
752347 
752642 

9.752937 
763231 
753626 
763820 
754115 
754409 
754703 
754997 
755291 
765685 

9.755878 
766172 
766466 
756759 
757062 
757345 
757638 
767931 
758224 
758517 

9.758810 
-759102 
759395 
759687 
759979 
760272 
760564 
760856 
761148 
761439 


D.  10' 


Cotang. 


49.6 
49.6 
49.6 
49.6 
49.6 
49.6 
49.5 
49.5 
49.5 
49.6 
49.5 
49.4 
49.4 
49.4 
49.4 
49.4 
49.3 
49.3 
49,3 
49.3 
49,3 
49.3 
49.2 
49.2 
49,2 
49.2 
49,2 
49,2 
49.1 
49.1 
49,1 
49,1 
49.1 
49,1 
49.0 
49,0 
49.0 
49.0 
49,0 
49.0 
48.9 
48.9 
48,9 
48,9 
48,9 
48,9 
48.8 
48.8 
48.8 
48.8 
48.8 
48.8 
48.7 
48.7 
48.7 
48.7 
48.7 
48.7 
48.6 
48.6 


10.256248!  48481 187462 
255950  i,485fX):87448 
265652  J  48532J87434 
255,3551,48557  87420 
255067!  148583:87406 


Cotang.   N.  HJne.iN.  COS. 


254760 
254462 
264165 
'253868 


1 48608187391 
4863487377 
4865987363 
48684'87349 


253671  '48710,87335 
253274  j  48735:87321 

10.252977  i  4876 li87 306 
252681  !  48786187292 
252384  14881187278 
252087  1 48837187264 
251791  148862:87250 
251496  148888  87235 
251199  l4891.3;87221 
260903  j  48938:87207 
250607  !  48964187193 
250311  j  48989  87178 

10.250015  1149014  87164 


249719 
249424 
249128 
248833 
248538 
248243 
247948 
247633 
247358 
10.247063 
246769 
246474 


49040;87150 
49065J87136 
49090,87121 
49116|87107 
4914187093 
49166i87079 
49192J87064 
49217187050 
49242J87036 
4926887021 
4929o87007 
4931886993 
2461801149344  86978 
245885114936986964 
245591 1 1 49394  86949 
245297  1 14941986935 
245003!;  49445  86921 
2447091,49470,86906 
244416  1149495  86892 
10.244122^14952186878 
243828 '149546  86863 
2436361  49571 186849 
243241  ji  49596  86834 
242948!  149622,86820 
242655  1 1 49647 186805 
242362  ij  49672  86791 
242069 1  49697186777 
241776  [1 49723186762 
2414831:49748  86748 
10.241190:  49;73  86733 
240898  49798  86719 
2406051.49824  86704 
240313  !|  49849  86690 
240021!!  49874186676 
2397281:49899,86661 
239436  ii  49924  86646 
239144  j  I  49960;86632 
238852 149975|86617 
238561  !  6000U  86603 


Tang.   Ii  N.  co>i.|N.f;ii 


Degrees. 


TABLK  IL 


Log.  Siaes  aaJ  Taagonta.    (30°)    Natural  Sines. 


51 


Sine.      D.  10' 


)  G98!i70!„p 

6M&2'6  ,!! 
(i9!)8-t4  ,  ::^ 

7Ui)jt)-2  ::^ 

70U-280 ;  ^,. 
7004981^^ 
700716'^^ 
700933  rz 
701151 ;3« 

1.701368  i^,. 
701583  I  :J^' 
70180-2  j^^' 
70-2019  q..  ■ 
702'236:^^- 
70-2462  •  Tf. 
70-2ti«9.^^ 
702885,^° 
7031011^^' 
703317   ^ 

1.703533  !:? 
703749  :;? 
703964  r° 
704179;:;? 
704395 !  .^2 
7046101:;? 
7048-25!^? 

705040;:;? 

705-254;:^? 
705469':;? 

1.705683 :;? 

705898  ^? 
706112  ^?- 

706326  :;?• 
706539  i:;?  • 

706  753  M?  • 
70696/!:^?- 

70/180  i:;?  • 
707393  i:;^ 

707606^5, 

1.707819  :;? 

708032  ^? 

708245  :;? 

708458  :;? 

70^6.0  :;? 

708882  :;? 

70J0J4  :;? 

70J30J  :;? 

709518  :;?• 
7097301^? 

709941  :;? 
71015J  :;? 

710>64  ^? 
710575  -^^ 
710786 
710J67 
711-208 
711419  r" 
7116-29  ^? 
711839  "^^ 


Cosine. 


Cosine. 

.937531 
93  7458 
937385 
937312 
937238 
937165 
937092 
937019 
936946 
936872 
936799 
.936725 
936()52 
936578 
936505 
936431 
936357 
936-284 
936210 
936136 
936062 

S.  935988 
935914 
935840 
935766 
935692 
935618 
935543 
935469 
935395 
935320 
,935-246 
935171 
935097 
935022 
934948 
934873 
934798 
9347-23 
934649 
934574 
934499 
934424 
934349 
934274 
934199 
9341-23 
9340  4-S 
9339  73 
933898 
933822 

9.933747 
933671 
933596 
933520 
933445 
933369 
933293 
933217 
933141 
933066 


Sine. 


D.  10' 


12.1 
12.2 
12.2 
12.2 
12.2 
12.2 
12.2 
12.2 
12.2 
12.2 
12.2 
12.2 
12.3 
12.3 
12.3 
12.3 
12.3 
12.3 
12.3 
12.3 
12.3 
12.3 
12.3 
12.3 
12.4 
12.4 
12.4 
12.4 
12.4 
12.4 
12.4 
12.4 
12.4 
12.4 
12.4 
12.4 
12.4 
12.5 
12.5 
12.5 
12.5 
12.5 
12.5 
12.5 
12.5 
12.5 
12.3 
12.5 
12.5 
12.6 
12.6 
12.6 
12.6 
12.6 
12.6 
12.6 
12.6 
12.6 
12.6 
12.6 


Tang. 

.761439 
761731 
762023 
76-2314 
762t»06 
76-2897 
763188 
763479 
763770 
764061 
764352 

.  764(i43 
764933 
766224 
765514 
765806 
766095 
766385 
766676 
766965 
767255 

. 767545 
767834 
768124 
768413 
768703 
768992 
769-281 
769570 
769860 
770148 

.770437 
7707-26 
771015 
771303 
771592 
771880 
772168 
77-2457 
772745 
773033 
773321 
773608 
773896 
774184 
774471 
774769 
775046 
775333 
775621 
775908 
'.776ly5 
776482 
776769 
777055 
777342 
77  76-28 
777915 
778201 
778487 
778774 

Colatig. 


9. 


D.  10" 


48.6 
48.6 
48.6 
48. «i 
48.5 
48.5 
48.5 
48.5 
48.5 
48.5 
48.4 
48.4 
48.4 
48.4 
48".  4 
48.4 
48.4 
48.3 
48.3 
48.3 
48.3 
48.3 
48.3 
48.2 
48.2 
48.2 
48.2 
48.2 
48.2 
48.1 
48.1 
48.1 
48.1 
48.1 
48.1 
48.1 
48.0 
48.0 
48.0 
48.0 
48.0 
48.0 
47.9 
47.9 
47.9 
47.9 
47.9 
47.9 
47.9 
47.8 
47.8 
47.8 
47.8 
47.8 
47.8 
47.8 
47.7 
47.7 
47.7 
47.7 


Cotiing. 

10.-238561 
238269 
23  7977 
237686 
237394 
237103 
236812 
236521 
236230 
235939 
235648 

10.235357 
2350J7 
234776 
234406 
234195 
233905 
2336 ' s 
2333.46 
233035 
232745 

10.23-2455 
232166 
231876 
231587 
231297 
231008 
230719 
230430 
230140 
2-29H52 

10-2-29563 
229274 
2-2-985 
2-28697 
228408 
228120 
227832 
227543 
227255 
2-26967 

10.2-26679 
2-26392 
2-26104 
2-25816 
225529 
2-25-241 
224954 
224667 
224379 
224092 

10-2-23805 
2-23618 
223231 
222945 
222658 
2-2-2372 
2-22085 
221799 
221612 
2212-26 


N.  siiio.  N.  cos 

8(5603 
86588 
■S6573 
8(i559 
86544 
86530 
8(i515 
.S6r,01 
8(i486 
86471 
86457 
86442 
86427 
86413 
8t)398 
86384 
8()369 
8{i354 
86340 
86325 
86310 
8ti295 
86-281 
86266 
86-251 
86-237 
86222 


; 60000 
600 
50051) 
60076 
60101 
50126 
50151 
50176 
5,J201 

,50-22/ 
50252 
50277 
60302 
50327 

1 60352 
50377 

1 150403 
60428 
60453 
60478 
60503 
56528 
60553 
60578 

|5(Xi03 

' 50628 

1 6mi54 


;  60079  86-207 
;  60704^192 


i  507-29 
60754 
160779 
: 50804 
1 50829 
60854 
: 60879 
i  50904 


86178 
86163 
86148 
86133 
86119 
86104 
86089 
86074 


■  50929  86059 
i  50954  86045 

509(9  86030 
'51004  86015 
,610-29  86000 
151054  85985 

51079  85970 


-i! 


61104 
611-29 
61154 
61179 


151-204 
61229 


rt5906 
85941 
859-26 
85911 
85896 
85881 


61254|85866 
61279185851 
6130^85836 
513-29  «5821 


Tang. 


161354 
51379 
51404 
51429 
'51451 
; 61479 
:fl504 
I  N.  CCS. 


85806 
85792 
85777 
85762 
86747 
85732 
8^717 
.N.gine 


59  iJejprees. 


21 


52 


Log.  Sines  and  Tangents.    (31°)    Natural  Sines. 


TABLE  I] 


0 
1 

2 
3 
4 
5 
6 
7 
8 
9 
lOl 

'M 

12 

13' 

14' 

15 

16 

17 

18 

19 

20 1 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

3b 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

60 

51 

62 

63 

64 

65 


Sine. |D.  10"|    Cosine. 

K711839L. 

7120501^? 

712260  J- 

712469   ^J] 

712679   tl 

71-2889  t; 

713098  t; 

713308  3^ 

713517   o^ 

713726  q^ 

713935  r,. 
)  714144  ^^ 

714352  ^T 

714561  t: 

714769  r.t 

714978  t; 

715186  tl 

715394  t: 

715602  t: 

715809  5;t 

716017  t: 

1.716224  ^2' 

716432  t: 

716639  ^7 

716846  I  ^7  ■ 

717053  i^^' 

717259^^ 

717466^^ 

717673  :5^ 

717879  :5^' 

718085  r.1' 
>. 718291  T: 

718497  ij;: 

718703'^;: 

718909  t: 

719114  t: 

719320  r.T 
71y525  "i^ 

719730  ::;: 

719935  "^T 

720140  ::]:■ 

1.720346  :^t' 

720549  '^T 

720754  ^J' 

720958  ^T 

721162  ^J' 

721366  ^J 

721570  ^l' 

721774  ^*' 

721978  ii 

722181  ^^ 
1.722385  ^^ 

722588  a 

•722191  a 

722994  ^5 

723197  t^ 

723400  ^1;' 

723603  r,i 

723805 -^t 

724007  ^l 
_7242_l0  ^ 

Ccsine. 


9.9330;)6 
932990 
932914 
932838 
932762 
932686 
932609 
932533 
932467 
932380 
932304 

9.932228 
932161 
932075 
931998 
931921 
931845 
931768 
931691 
931614 
931537 

9.931460 
931383 
931306 
931229 
931152 
931076 
930998 
930921 
930843 
930766 

9.93(J688 
930611 
930533 
930466 
930378 
930300 
930223 
930145 
930067 
929989 
.929911 
929833 
929756 
-  929677 
929699 
929521 
929442 
929364 
929286 
929207 
.929129 
929060 
928972 
928893 
928816 
928736 
928657 
928578 
928499 
928420 
~Siue.~ 


D.  10' 


12.6 
12.7 
12.7 
12.7 
12.7 
12.7 
12.7 
12.7 
12.7 
12.7 
12.7 
12.7 
12.7 
12.8 
12.8 
12.8 
12.8 
12.8 
12.8 
12.8 
12.8 
12.8 
12.8 
12.8 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12,9 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 


Tang.   D.  10" 


778774 
7790a0 
779346 
779632 
779918 
780203^ 
780489- 
780775 
781060 
781346 
781631 

9.781916 
782201 
782486 
782771 
783066 
783341 
783626 
783910 
784195 
784479 
784764 
785048 
785332 
785616 
785900 
786184 
786468 
786762 
787036 
787319 

9.787603 
787886 
788170 
788453 
788736 
789019 
789302 
789585 
789868 
790151 
.790433 
790716 
790999 
791281 
791663 
791846 
,  792128 
792410 
792692 
792974 
1.793266 
793638 
793819 
794101 
794383 
794664 
794946 
795227 
795508 
795-789 
Cotaug. 


47.7 
47.7 
47.6 
47.6 
47.6 
47.6 
47.6 
47.6 
47.6 
47.6 
47.6 
47.5 
47.5 
47.5 
47.6 
47.6 
47.6 
47.4 
47.4 
47.4 
47.4 
47.4 
47.4 
47.3 
47.3 
47.3 
47,3 
47.3 
47.3 
47.3 
47.2 
47.2 
47.2 
47.2 
47.2 
47.2 
47.2 
47.1 
47.1 
47.1 
47.1 
47.1 
47.1 
47.1 
47.1 
47.0 
47.0 
47.0 
47.0 
47.0 
47.0 
47.0 
46.9 
46.9 
46.9 
46.9 
46.9 
46.9 
46.9 
46.8 


^u«-     I 


Cotaug.     iiN.sine.lN.  cos. 


161604186717 
I  5 1629186  7  02 
51554185687 
61679185672 
'  61604!86667 
;  61628185642 
61653185627 
I51678I861J12 
:  61703 185597 
'61 728 '85582 
61763 '85567 
161778185551 
i  51803185636 
i  5182b  18552 1 
151852185506 
!  61877186491 
I  61902185476 
,  51927 186461 
l!  51952185446 
,;61977!8o43l 
!  62002185416 
P  62026i864(il 
il6205ll853S5 
i62076!86370 
;  62101185356 
62126JH6340 
,521511853261 
!  62175!863lOi 
i;62200i8o294| 
]i62226|S62v9! 
1 62250|852o4  | 
1 62275i86249  I 
1 62299185234 
!62324|85218 
j  52349  85203  I 
152374185188  1 
|62a99jh5l73| 
i624--;3!85l67i 
i  52448ib5l42' 
i  6247  3185127 
;  62498i85112 
j;  62522185096 
'i  52647185081 
I  6'-<;57".i  85006 
|:  6259/ 185051 
I  62(j21|850ii5 


10.221226 
220940 
220654 
2203^68 
220082 
219  797 
219611 
219225 
218940 
218654 
218369 

10.218084 
217799 
217514 
217229 
216944 
216659 
216374 
216090 
215805 
215521 

10.216236 
214952 
214668 
214384 
214100 
213816 
213632 
213248 
212964 
212681 

10.212397 
212114 
211830 
211547 
2 11264 
210981 
210by8 
210415 
210132 
209849 

10.209567 
209284 
209001 
208719 
208437 
208154 
207872 
207  590 
207308 
207026 

10.206744 
206462 
206181 
205899 
205617 
205336 
205055 
204773 
204492 
204211 


52646185020 
5267lib5«»«>5 
52()96i849b9 
62720j84974 
52745i849j9 
6277<»l84943 
627941849-8 
52biy[b49l3 
62b44!b4by7 
528o9l84b82 
52b;'oi84b06 
5-^918J84b5l 
62943  i84»3t) 
62967i84b20 
52992184805 


Tang.       ''  ^^  co.>^.|n  sine. 


:J 


68  Degrees. 


TABUS  II. 


Log.  Sines  iiud  Tangents.    {3'2P)    Natural  Sines. 


53 


Sine. 

9.724210 
724412 
724  j1  4 
724811) 
725U17 
725219 
725420 
725u22 
725823 
72G024 
72()225 
720421) 
720ii20 
720827 
727027 
727228 
727428 
727028 
727828 
728027 
728227 

9. 72842/ 
728020 
728825- 
729024 
729223 
729422 
729021 
729820 
730018 
730210 

9.730415 
730013 
730811 
731009 
731200 
731404 
731002 
731799 
731990 
732193 

9.732390 
732587 
732784 
732980 
733177 
733373 
733509 
733705 
733901 
734157 

|9. 734353 
734549 
734744 
734939 
735135 
735330 
735525 
735719 
735914 
73()109 
Cosine. 


D.  10"!    Cosine.    |D.  10"[     Titng 


33,7 

33.7 
33.6 
33.6 
33.6 
33.6 
33.5 
33.5 
33.5 
33.5 
33.5 
33.4 
33.4 


33.4 
33.4 
33.4 
33.3 
33.3 
33.3 
33.3 
33.3 
33.2 
33.2 
33.2 
33.2 
33.1 
33.1 
33.1 
33.1 
33.0 
33.0 
33.0 
33.0 
33.0 
32.9 
32.9 
32.9 
32.9 
32.9 
32.8 
32.8 
32.8 
32.8 
32.8 
32.7 
32.7 
32.7 
32.7 
32.7 
32.6 
32.6 
32.6 
32.6 
32.5 
32.5 
32.5 
32.5 
32.5 
32.4 
32.4 


9.928420 
9283 12 
928203 
928183 
928104 
928025 
927946 
927807 
927787 
92 7 7 OS 
927629 

9.927549 
927470 
927390 
927310 
927231 
927151 
927071 
92(i991 
92091 1 
920831 

9.920751 
920()71 
920591 
92051 1 
926431 
920351 
920270 
920190 
920110 
920029 

9.925949^ 
925808 
925788 
925707 
925020 
925545 
926405 
925384 
-925303 
925222 

9.925141 
925000 
924979 
924897 
924816 
924735 
924654 
924572 
924491 
924409 

9.924328 
924246 
924164 
924083 
924001 
923919 
923837 
923755 
923073 
923591 
Sine. 


13.2 
13.2 
13.2 
13.2 
13.2 
13.2 
13.2 
13.2 
13.2 
13.2 
13.2 
13.2 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 


13 

13 

13 

13 

13 

13 

13 

13 

13 

13.5 

13.6 

13.6 

13.6 

13.6 

13.6 

13.6 

13.6 

13.6 

13.6 

13.6 

13.6 

13.6 

13.7 

13.7 


9. 


,795789 
79ii0T0 

790032 
79ti913 
797194 
797475 
797755 
798036 
798316 
798596 
,798877 
799157 
799437 
799717 
799997 
800277 
800557 
800836 
8((1116 
801396 
801076 
801955 
802234 
802513 
802792 
803072 
803351 
803(,30 
803908 
804187 

>.8044<)6 
804745 
805023 
805302 
805580 
805859 
800137 
806415 
800093 
800971 

>.  807249 
80.627 
807805 
80;-5083 
808361 
80:-^638 
808916 
8(»9193 
809471 
809748 

>.810(»25 
810302 
810580 
810857 
811134 
811410 
811687 
811964 
812241 
812517 


D.  10"l  CoUmg.   N.  sine.jN.  I'O.^. 

10.204211  i  52992184805 
203:^30  63017  84789 
203(i49  53041;b4774 
203308'  63000|84759 
203087;  63U9 184743 
202800!  53115184728 
2u2525i  63140'84712 
202245  63164  84(i97 
201904  ,63189  84081 
2016841  63214184660 
2014041  63238!84()50 

10.201123;  63203184035 
200843  ,63288  84019 
200563  ;  6331 2i84()04 
2002831  63337184588 
200003]  63301:84573 
199723  6338():84557 
199443  1  6341  ri84542 
199104  I  6343584520 
1988841  63400  84511 
198004  63484  84495 

10.198325  1  53509  84480 
198045'  63534  84404 
197766  j:  53558  8444S 
197487  '  53583  84433 
197208  53(i-j:  84417 
190928  I  53032  H4402 
190049  I  63C;50  84280 
1963701  63681  84370 
1900921!  63705  84355 
195813;  63730  84339 

10.195634  5376484324 
195256  63779  84308 
194977  163804  84292 
194098  6382884277 
194420  63853-84201 
194141  63877  84245 
193803  '53902  84230 
193585  63920  84214 
193307-6395184198 
193029.  63975  84182 

10.192751'  64000  84107 
192473  '54024  84151 
192195 '64049  84135 
191917  116407384120 
191039!  6409  i  84104 
191302  I' 64122  84088 
191084,  6414«>  84072 
190807  i  6417184057 
1905291  64195  84041 
1902621164220  84025 

10. 189976;!  64244  84009 
189098  I  64209,83994 
189420  64293*83978 
189143  ,'64317  83962 
l888<)6li543428o946 
188590;' 64300  83930 
188313  64;-9r8o915 
188030;  64415 8o 899 
187769  :,6444U 83883 
187483  I  64404,83807 
Tang.   ,  N.  ros.|N.sinc, 


57  Degrt^B. 


r 


54 


Log.  Sines  and  Tangents.    (330)    Natural  Bines.  TABLE  II. 


9. 


Sine. 

736303 
736498 
736692 
73t)886 
737080 
737274 
737467 
737661 
737855 
738048 
.738241 
738434 
738627 
738820 
739013 
739206 
739398 
739590 
739783 
739976 

.740167 
740359 
740550 
740742 
740934 
741125 
741316 
741508 
741699 
741889 

.742080 
742271 
742462 
742652 
742842 
743033- 
743223 
743413 
743602 
743792 
743982 

-744171 
744361 
744550 
744739 
744928 
745117 
745306 
745494 
745683 
7458/1 
746059 
746248 
746436 
746624 
746812 
74699y' 
747187 
747374 
747562 

Cosine. 


D.  10" 


32,4 
32.4 
32.4 
32.3 
32.3 
32.3 
32.3 
32.3 
32.2 
32.2 
32.2 
32.2 
32.2 
32.1 
32.1 
32.1 
32.1 
32.1 
32.0 
32.0 
32.0 
32.0 
32.0 
31.9 
31.9 
31.9 
31.9 
31.9 
31.8 
31.8 
31.8 
31.8 
31.8 

31;7 

31.7 
31.7 
31.7 
31.7 
31.6 
31.6 
31.6 
31.6 
31.6 
31.5 
31.5 
31.5 


Cosine. 


31 

31 

31 

31 

31 

31 

31.4 

31.3 

31.3 

31.3 

.31.3 

31.3 

31.2 

31.2 


,923591 
923509 
923427 
923345 
923263 
923181 
923098 
923016 
922933 
922851 
922768 
922686 
922603 
922520 
922438 
922355 
922272 
922189 

I  922106 
922023 
921940 

9.921857 
921774 
921691 
921607 
921524 
921441 
921357 
921274 
921190 
921107 

9\ 92 1023 
920939 
920856 
920772 
920688 
920604 
920520 
920436 
920352 
920268 

9.920184 
920099 
920015 
919931 

"  91^846 
919762 
919677 
919593 
919508 
919424 

9.919339^ 
919254 
919169 
919085 
919000 
918915 
9188.i0 
918745 
918669 
918574 
~Sine7~ 


D.  10' 


13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.0 

13.9 

13.9 

13.9 

13.9 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.2 

14.2 

14.2 

14.2 


Tung. 


9.812517 
812794 
813070 
813347 
813623 
813899 
814175 
814452 
814728 
815004 
816279 
9.815555 
815831 
816107 
816382 
816(J58 
816933 
817209 
817484 
817759 
818035 
818310 
818585 
818860 
819135 
819410 
819684 
819959 
820234 
820508 
820783 
9.821057 
821332 
821606 
821880 
822154 
822429 
822703 
822977 
823250 
823524 
9.823798 
824072 
824345 
824619 
824893 
825166 
825439 
825713 
825986 
826259 
9,826532 
826805 
827078 
827351 
827624 
82  7897 
828170 
828412 
828  716 
82898  7 
C^^oUin'T. 


D.  10 


46.1 
46.1 
46  1 
46.0 
46.0 
46.0 
46.0 
46.0 
46.0 
46.0 
46.0 
4 1.9 
45.9 
45.9 
45.9 
45.9 
45.9 
45.9 
45.9 
45.9 
45.8 
45.8 
45.8 
45.8 
45.8 
45.8 
45.8 
45.8 
45.8 
45.7 
45.7 
45.7 
45.7 
45.7 


Cotang.  j  N.  .sine.lN.  cos. 


10, 


10 


187482 

187206 

186930 

186653  i 

186377 j 

186101 : 

185825 

185548, 

185272 

184996 ' 

184721 

184445 

184169 

183893: 

183618: 

183342  i 

183067 : 

182791 : 


5446483867 
64488  83851 
64513  83835 
64537  83819 
6456183804 
54586183788 
64610183772 
54635,83756 
54659  83740 
54683  83724 
54708:83708 
64'}  32 ,83692 
54756  83676 
54781  83660 
54805  83645 
C482y;83629 
54854:83613 
54878  83597 


45 

46 

45 

45 

45 

45 

46 

46 

45 

45.6 

45.6 

45.6 

46.6 

46.5 

45.5 

45.6 

45.6 

45.6 

45.6 

45.6 

45.5 

45.5 

46.4 

45.4 

45.4 

45.4 


182516:  54902  83581 
18224]  54927  83565 
181965 '  54951:83549 

10.181690  64975  83533 
181415  54999  83517 
181140  560^4  83501 
180865  55048  83486 
180590  55072  83469 
180316  55097  83453 
180041  5512183437 
179/66  65145  83421 
179492  55169  83405 
179217  !i  55194  83389 

10.178943:  55218  83373 
178668  55-42  83356 
178394!  55266  83340 
178120:  55^9183324 
177846:  55315  83308 
177671  j  55339  83292 
177297!;  55363183276 
177023  i  56388  83260 
176760  i  1 55412  83244 
176476 1 155436 '83228 

10. 176202  ji  65460  83212 
176928  i|55484!83195 
175655  ji  65509  83179 
176381s  65533  j83163 
176107]:  65557183147 
1748341!  55581 183131 
174561  155605  83115 
174287 1!  55630  83098 
1740141'!  5566 1  83082 
173741  i  k.5678  830ij6 


10.1734681 
1731951 
172922  I 
172649'! 
172376 
172103  |i 
171830: 
171558 
171-.^85 
171013 
Tang. 


55702  83060 
55726  83034 
55760  83017 
5577583U01 
55799  82L85 
65823  821.69 
:  55847  :82*j53 
j  55871  82036 
j  5.5895  ;82!:>20 
55^19  82904 


l_ 
'60 
59 
'58 
57 
56 
56 
54 
53 
52 
51 
50 
49 
148 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
16 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
<) 

T 

0 


56  Degrees. 


Log.  Sines  and  Tangents.  (34°)  Natural  Sines. 


55 


M  _Sint 

019  7475G2 

1 

2 

3 

4 

6 


7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
4i. 
41 
42 
43 
44 
45 
Mi 
47 
48 
49 
50 
51 
52 
53 
64 
55 
56 
57 
68 
59 
60 


747749 
74793!) 
748123 
748310 
7'1849  7 
74H683 
748870 
749056 
749243 
749426 

9.74!>()15 
749801 
749987 
750172 
750358 
760543 
750729 
750914 
751099 
751284 

9.751469 
751654 
751839 
752023 
7522(W 
752392 
752576 
752760 
752944 
753128 

9.753312 
753495 
753679 
7638()2 
754046 
754229 
754412 
754595 
754778 
754960 
,755143 
755326 
755508 
755()y0 
75r>872 
756054 
756236 
756418 
756(iO0 
756782 

9.756y()3 
757144 
75732(> 
757507 
7576.S8 
7578()9 
758050 
768230 
758411 
758591 
Cosine. 


D.  10' 


31.2 


31 

31 

31 

31 

31 

31 

31 

31 

31.0 

31.0 

31.0 

31.0 

30.9 

30.9 

30.9 

30.9 

30.9 

30.8 

30.8 

30.8 

30.8 

30.8 

30.8 

30.7 

30.7 

30.7 


30 

30 

30 

30 

30 

30 

30 

30 

30 

30.5 

30  5 

30.5 


30 

30 

30 

30 

30 

30 

30 

30 

30.3 

30.3 

30.3 

30.2 

30.2 

30.2 

30.2 

30.2 

30.1 

30.1 

30.1 

30.1 

30.1 


Co.«ine. 

>.  9 18574 
918480 
918404 
918318 
918233 
918147 
9180)2 
917976 
917891 
-917805 
917719 
1.917634 
917548 
917462 
917376 
917290 
917204 
917118 
917032 
916046 
916859 
.916773 
916687 
916600 
916514 
916427 
916341 
916264 
916167 
916081 
915994 
.915907 
915820 
915733 
915646 
915559 
915472 
915386 
916297 
915210 
915123 
.915035 
914948 
914860 
914773 
914685 
914598 
914510 
914422 
914334 
914246 
.914168 
9140i0 
913982 
913894 
913806 
913718 
913630 
913541 
913453 
913365 


D.  10" 


14.2 

14.2 

14.2 

14.2 

14.2 

14.2 

14.2 

14.3 

14.3 

14.3 

14.3 

14.3 

14.3 

14.3 

14.3 

14.3 

14 

14 

14 


14 

14 

14 

14 

14 

14 

14 

14 

14 

14 

14 

14.6 

14.5 

14.6 

14.5 

14.6 

14.6 

14.5 

14.6 

14.5 

14.5 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.7 

14.7 

14.7 

14 


14.7 


"  Tang. 

9.828987 
82!i260 
829532 
829805 
830iJ77 
830349 
830o2l 
830893 
831165 
831437 
831709 

?. 831981 
832253 
832525 
832796 
833068 
833339 
833611 
833882 
834164 
834425 

). 834696 
834967 
835238 
835509 
835780 
836051 
836322 
836593 
836864 
837134 

). 837405 
837676 
837946 
838216 
838487 
838767 
839027 
83.9297 
839568 
839838 

1.840108 
840378 
840647 
840917 
841187 
841467 
841726 
84199<) 
842266 
842536 

1.842806 
843074 
843343 
843612 
843882 
844151 
844420 
844689 
8449.^)8 
845227 
Cotang. 
65  Degrees. 


D.  10" 


45 

45 

45 

45 

45 

45 

45 

45 

45 

45.3 

45.3 

45.3 

45.3 

45.3 

45.3 

45.2 

45.2 

45 

45 

45 

45 

45 

45 

45 

45.2 

45.1 

45.1 

45.1 

45.1 

45.1 

45.1 

45.1 

45.1 

45.1 

45.1 

45.0 

45.0 

45.0 

45.0 

46.0 

45.0 

45.0 

45.0 

45.0 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.8 

44.8 

44.8 

44.8 

44.8 


Cotang.  I  A. sine 


10 


1710131 

170740 

1704()8' 

170195! 

169923 ! 

169651 

169379 


65919 
55943 
559o8 


.N.  COS. 


S2904 

H2KH7 
82871 


I  56064 
169107:156088 


55992  82855 
56016182839 
56040  82822 
82806 


82790 
82773 
8275' 


168835  !|  561 12 

168563  ij  56136, 

lti829li|  66160  82741 
10.1680191156184  82724 


167747 
167476 
167204 
166932 
166661 
166389 
166118 
166846 
165576 
10.165304 
166033 
164762 


6(i208 
56232 
56256 
56280 
56305 
i  56329 
' 56353 
I  56377 
156401 
t  56425 
56449 
56473 


164491  1 1 66497 
164220  I  j 66521 
1639491156545 
163678!;  56569 


163407 
163136 
162866 
10.162595 
162325 
162054 
1617841:56736 
161613  ji  66760 
161243  1  56784 


66593 
'56617 

56641 
j 56665 

66689 

56713 


82708 
82()92 
82675 
82(i59 
82»i43 
82()26 
82610 
82593 
82577 
82561 
82544 
82528 
82511 
82495 
82478 
82462 
82446 
82429 
82413 
82396 
82380 
82363 
82347 
82330 
82314 


160973  1156808  8229 


160703  il568b2 
160432  156856 
160162  166880 
10.159892  Ij  56904 
159622  I  56928 
159353  1 56962 
159083!  156976 
158813;  I  670U0 
158543 
158274 
158004 
167734 
167465 
157195 
156926 
156657 
166388 


10 


57024 
57047 
57071 
57095 
57119 
67143 
67167 
57191 
57215 


82281 
82264 
82248 
82231 
82214 
82198 
82l8l! 16 
821651 15 
82148' 14 
82132! 13 
82115; 12 
82098,11 
820821  10 


15611811 67238 
1558491  57262 
155580;!  67286 
155311  157310 


165042 
154773 


Tnng. 


57334 
67368 


82065 
82048 
H2032 
82015 
81999 
81982 
81966 
81949 
81932 
81916 


9 
S 
7 
6 
6 
4 
3 
2 

6|  0 


N.  COS.  N  sine. 


J 


56 


Log.  Sines  and  Tangents.    (35°)    Natural  Smes. 


TAIJLK  II. 


0 
1 

2 
3 
4 
5 

n 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
26 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
3S 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
66 
67 
68 
59 
60 


9. 


75S591 
758772 
758952 
759132 
759312 
759492 
759d72 
759852 
760031 
760211 
760390 
76^569 
760748 
7b0927 
761106 
761285 
761464 
761642 
761821 
761999 ! 
762177! 
762356  j 
762534  I 
7627121 
762889  i 
763067  i 
763245  i 
763422 
763600 
763777 
763954 
764131  I 
764308 
764485 
764662 
764838 
765015 
765191 
765367 
765544 
765720 
765896 
766072 
766247 
766423 
766598 
766774 
766949 
767124 
767300 
767475 
.767649 
767824 
767999 
768173 
768348 
768522 
768697 
768871 
769045 
769219 
Cosine. 


1).  lU' 

30.1 
30.0 
30. U 
30  0 
30. u 
30.0 
29.9 
29.9 
29.9 
29.9 
29.9 
29.8 
29.8 
29.8 
29.8 
29.8 
29.8 
29.7 
29.7 
29.7 
29.7 
29.7 
29.6 
29.6 
29.6 
29.6 
29.6 
29.6 
29.5 
29.5 
29.5 
29.5 
29.5 
29.4 
29.4 
29.4 
29.4 
29.4 
29.4 
29.3 
29.3 
29.3 
29.3 
29.3 
29.3 
29.2 
29.2 
29.2 
29.2 
29.2 
29.1 
29.1 
29.1 
29.1 
29.1 
29.0 
29.0 
29.0 
29.0 
29.0 


Cosine. 

1.913365 
913276 
913187 
913099 
913010 
912922 
912833 
912744 
912655 
912566 
912477 

1.912388 
912299 
912210 
912121 
912031 
911942 
911853 
911763 
911674 
911584 

1.911495 
911405 
911315 
911226 
911136 
911046 
910956 
910866 
910776 
910J86 

.91U596 
910506 
910415 
910325 
910236 
910144 
910054 
909963 
909873 
909782 

.909691 
909501 
909510 
909419 
909328 
90923  7 
909146 
909055 
908964 
90>"i873 

1.908781 
908690 
9085,9 
908507 
908416 
908324 
908233 
908141 
908049 
907958 
Sine. 


L>.  iU 


4.7 
4.7 
4.8 
4.8 
4.8 
4.8 
4.8 
4.8 
4.8 
4.8 
4.8 
4.8 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4,9 
4.9 
5.0 
5.0 
5.0 
5.0 
5.0 
5.0 
5.0 
5.0 
5.0 
5.0 
5.0 
5.1 
5.1 
5.1 
5.1 
5.1 
5.1 
5.1 
5.1 
5.1 
5.1 
5.1 
5.2 
5.2 
5.2 
5.2 
5.2 
5.2 
5.2 
5.2 
5.2 
5.2 
5.3 
5.3 
5.3 
5.3 


T:iii>j.  D.   10 


9.845227 
84549.) 
845  764 
846033 
846302 
846570 
846839 
847107 
847376 
847644 
847913 

.848181 
848449 
848717 
848986 
849254 
849522 
849790 
850058 
850325 
850593 

.850851 
851129 
851396 
851664 
851931 
852199 
852466 
852733 
853001 
853268 

.853635 
8538i)2 
854069 
854336 
854603 
854870 
855137 
855404 
865671 
855938 

.856204 
856471 
856737 
857004 

^57270 
857537 
867803 
868069 
858336 
868602 
9.868858 
859134 
859400 
859666 
859932 
860198 
860464 
86J730 
86J995 
861261 

Cotiin'T. 


44.8 

44.8 

44.8 

44.8 

44.8 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.6 

44.6 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44.5 

44.5 

44.5 

44.6 

44.6 

44.5 

44.5 

44.5 

44.5 

44.5 

44.4 

44.4 

44.4 

44.4 

44.4 

44.4 

44.4 

44.4 

44.4 

44.4 

44.4 

44.3 

44.3 

44.3 

44.3 

44.3 

44.3 

44.3 

44.3 

44.3 

44.3 


10, 


10 


10 


10 


10 


10 


154; 73 
154504 
154236 
153967 
153698 
153430 
153161 
152893 
152(i24 
152356 
15C087 
151819 
151551 
151283 
151014 
150746 
150478 
150210 
149942 
149675 
149407 
149139 
148871 
148604 
148336 
148059 
147801 
147534 
147267 
146999 
146732 
146465 
146198 
145931 
145664 
145397 
146130 
144863 
144596 
14432y 
144062 
143796 
143529 
143263 
142996 
142730 
142463 
142197 
141931 
141664 
141398 
141132 
140866 
140500 
140334 
140068 
139802 
139536 
139270 
1390J5 
138739 


I  N.  sine. IN.  cos. I 

;.  57358  81915  60 
'  57381 181899  59 
I  57 405 181  b&2  158 
I  57429.8 186{> 
||57453J81a48 
1 157477181832 
1!67'^01J81815 
I' 57624181  / 98  53 
ji57648|81782|52 
I  i  67572,81 765  '51 
|!57596i8174e 
'l57619|81731 
i  67643!81714 
!' 67667 I81C98 
!' 57691181681 
i  57715181664 
i:57738!81647 
ii  57762181631 
!' 67786  81614 
1:5781081597 

57833  81580 
i  57857  81563 
i:  67881  81546 
^57904  81530 
!  57928  81513 
!' 57952  81496 

57976  81479 
167999  81462 
j  58023  81445 
158047:81428 
i580/0;81412 
I  58094  8 1395 
i58118'813V8 
158141181361 
:68165|81344 

58189|81327 
,58'212|81310 
I  68236  [8 1293 
158260  81276 
j  58283 '8 1259 
158307181242 
I  5833081226 
'58354|81208 
'58378:81191 

68401 181 174 
;58425'811c7 


58449 
58472 
58496 


81140 
81123 
81105 i  12 
11 


i  5851981089 
I  58543  810^2 
58667 181055 
I  58590  8 1058 


1 158614 

I  58637 
i  158661 
i:  68684 
j' 58708 


81021 
81004 
80987 
80Ji0 
80953 


!58731|80j36 
i58755|80919 
i  58779,80902 


N.  COS.  N. sine. 


54  Degrees. 


TAIJLi:  II. 


Log.  Siiii'B  and  Tangents,    (36°)     NatKral  Sines. 


57 


D.  10" 


0 

9.7t)9219 

1 

769393 

2 

7(i9ot)ti 

3 

709740 

4 

769JJ13 

6 

77008/ 

6 

77t)2(>0 

7 

770433 

8 

770*  UK) 

y 

770779 

10 

7701);Vi 

11 

9.771125 

12 

771298 

13 

771470 

14 

771043 

15 

771815 

16 

771987 

17 

772159 

18 

-  772331 

19 

772503 

20 

772675 

21 

9.77284  7 

22 

773018 

23 

773190 

24 

773361 

25 

773533 

2t) 

773704 

27 

773875 

28 

774046 

29 

774217 

30 

774388 

31 

9.774558 

32 

774729 

33 

774899 

34 

775070 

35 

775240 

3(> 

775410 

3? 

775580 

38 

775750 

39 

775920 

40 

776090 

41 

9.776259 

42 

776429 

43 

776598 

44 

7767()8 

45 

776937 

-Hi 

77710;> 

47 

777275 

48 

777444 

49 

777613 

50 

777781 

51 

1).  7  7  7950 

52 

7:8119 

53 

778287 

54 

778455 

55 

778624 

56 

778792 

57 

7789vi0 

5S 

779128 

59 

779295 

GO 

779463 

29.0 

28.9 

28.9 

28.9 

28.9 

28.9 

28.8 

28.8 

28.8 

28.8 

28.8 

28.8 

28.7 

28.7 

28.7 

28.7 

28.7 

28.7 

28.6 

28.6 

28.6 

28.6 

28.6 

28.6 

28.6 

28.5 

28.5 

28.5 

28.5 

28.5 

28.4 

28.4 

28.4 

28.4 

28.4 

-.8.4 

28.3 

28.3 

28.3 

28.3 

28.3 

28.3 

28.2 

28.2 

28.2 

28.2 

28.2 

28.1 

28.1 

28.1 

28.1 

28.1 

28.1 

28.0 

28.0 

28.0 

28.0  I 

28.0 

28  0 

27  9 


Co.sine. 

.907958 
90;8(;6 
907774 
90?(i82 
907590 
907498 
907401) 
907314 
907222 
907129 
907037 

1.906945 
90()852 
906760 
906667 
906575 
90()482 
906389 
90()296 
906204 
906111 

1.906018 
905925 
905832 
905739 
905646 
905552 
905459 
905366 
905272 
905179 

1.905085 
904992 
904898 
904804 
904711 
904G17 
904523 
904429 
904335 
904241 

). 904 147 
904053 
903959 
9.)3S64 
903770 
903676 
903581 
903487 
903392 
903298 

i. 903202 
903108 
9J3014 
902919 
902824 
902729 
902634 
902539 
902444 
902349 


D.  10' 


Sine. 


15.3 

15.3 

15.3 

15.3 

15.3 

15.3 

15.3 

15.4 

15.4 

15.4 

15.4 

16.4 

16.4 

15.4 

15.4 

16.4 

15.4 

15.6 

16.6 

15.5 

15.6 

15.6 

15.6 

15  ' 

15 

15 

15 

16 

16 

1  ) 

15 

15 

15.6 

15.6 

15.6 

15.6 

16.6 

15.6 

15.7 

15.7 

15.7 

15.7 

15.7 

16.7 

16.7 

15.7 

15.7 

15.7 

16.7 

16.8 

16.8 

16.8 

16.8 

16.8 

15.8 

15.8 

16.8 

16.8 

16.9 

15.9 


9. 


Tang. 

861261 
861527 
861792 
8(i205S 
8(i2323 
862589 
862854 
863119 
863385 
8()3650 
863915 

.864180 
864445 
864710 
864975 
8t)5240 
865505 
865770 
866035 
866300 
86()5u4 

.866829 
867094 
867358 
867623 
867887 
868152 
8684 1^ 
868680 
868945 
869209 

.869473 
86973  7 
870001 
870265 
870529 
870793 
871057 
871321 
871685 
871849 

.872112 
872376 
8/2640 
872903 
873167 
873430 
873694 
873957 
874220 
874484 

1.874747 
875010 
875273 
875536 
h.5800 
876063 
876326 
876689 
876861 
877114 
Cotajig. 


D.  10"|  Cotaiig.   N.  sine.jN.  CU8 


44.3 
44.3 
44  2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44  2 
44.2 
44  1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.8 
43.8 
43.8 
43.8 
43.8 
43.8 
43.8 


10. 


10 


10 


10 


10 


10 


138739 
138473 
138208 
137942 
137677 
137411 
137146 
136881 ! 
136615 
136350 
13(i086 
135820 
1 35555 
136290 
135025 
134760; 
134495 
134230 
1339()5 
133700 
133436' 
133171 1 
1326061 
132642 j 
132377 
132113! 
131848; 
131584 
131320' 
131055 
130/91 
130527- 
13026311 
129999: 
129735 
129471' 
129207  ' 
128943 1  ■ 
128679 i 
128415! 
128151  I 
,  1278881, 
127624  I' 
12:360  : 
127097  I 
126833: 
126570' 
126306  I 
1260431 
125780' 
125516; 
.125253 
124990. 
124727 
124464 : 
124200,: 
123937 
123674  I 
123411' 
123149! 
122886' 

'ring.    !; 


68779  80902 
58802.80885 
68826  808()7 
68849  80850 
68873  80h33 
58896  80F 16 
68920  80799 
58943  80782 
58967  80765 
58990  80748 
59014  80730 
59037  80713 
69061  80()96 
6y(k*^4H(Kj7y 
69  lOO  80662 
69131  80644 
69154  80627 
69178  80()10 
59201  80593 
59225  80576 
59248  80568 
69272  80541 
69295  80524 
5931880507 
69342  80489 
59365  80472 
69389  80455 
59412  80438 
59436  80422 
59459  80403 
59-482  80386 
59506  80368 
59529  80361 
59552  80334 
6j5/6b03l6 
59599  80299 
69622  80282 
69640  80264 
696()9:80247 
69693  |802o0 
59;1()80-j1--^ 
59739  80 196 
59(63:801 1 8 
59786!80160 
59809  80143 
69832  8U 1 25 
59856  80108 
69879  80091 
69902  !800  73 
69926 '80051) 
59949  80038 
5!:»9;2  80021 
59995  800"3 
60019  79986 
60042 1/9968 
60065  i79961 
60089 17  9934 
60I12J/9916 
601351:9899 
60158i/9f)81 
6018279864 


N. 


to?., 


A' .sine 


53  Degn^fl. 


58 


Log.  Sines  and  Tangents.    (37°)    Natural  Sinos. 


TABLE  II. 


0 
1 

2 

3 

4 

5 

6 

~7 

8 

9 

\0 

11 

12 

13 

H 
15 
l(j 
17 
18 
19 
20 
21 
22 
23 
24 
25 
2t> 
27 
28 
29 
80 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
61 
52 
53 
54 
55 
56 
67 
58 
59 
60 


Sine.   D.  10' 


9.779463 

779o31 
779798 
7799o6 
780133 
780300 
780457 
7801)34 
780801 
780£>o8 
781134 

9.781301 
781468 
781634 
781800 
78 19b 6 
782132 
782298 
782464 
782630 
782796 
1.782961 
783127 
783292 
783458 
783623 
783/88 
783t»53 
784118 
784282 
-78444/ 

9.784612 
784776 
784941 
785105 
785269 
785433 
785597 
785761 
785925 
786089 

9.786252 
786416 
7865/9 
786  742 
786906 
7870o9 
78.232 
787395 
78755/ 
787720 
787883 
788045 


27.9 
27.9 
27.9 
27.9 
27.9 
27.8 
27.8 
27.8 
27.8 
27.8 
27.8 
27.7 
27.7 
27.7 
27.7 
27.7 
27.7 
27.6 
27.6 
27.6 
27.6 
27.6 
27.6 
27.  c 
27.5 
27.5 
2  7.6 
27.5 
27.5 
27.4 
27.4 
27.4 
27.4 
27.4 
27.4 
27.3 
27.3 
27.3 
27.3 
27.3 
27.3 
27.2 
27.2 
27.2 
27.2 
27.2 
27.2 


Cosine.  |D.  10' 


27.1 
27.1 
27.1 
27.1 
27.1 
27.1 


788208, 
788370 '"^'-^ 


788532 
788694 
788856 
789018 
789180 
789342 
I  Ccsine. 


27.0 
27.0 
27.0 
27.0 
27.0 
27.0 


902349 

902253 

902158 

902063 

90196  7 

901872 

901776 

901681 

901585 

9014^0 

901394 

901298 

901202 

901106 

901010 

900914 

900818 

900722 

900626 

900529 

900433 

900337 

900242 

900144, 

900047 

899951 

899854 

899757 

899660 

899564 

899467 

8993/0 

8992 /3H 

899176 

899078 

898981 

898884 

898787 

898689 

898592 

898494 

9.898397 
898299 
898202 
898104 
898006 
897908 
89/810 
"897712 
897614 
897516 

9.897418 
89/320 
897222 
897123 
89/025 

-896926 
896828 
896/29 
89v><.)31 
896532 


15.9 
15.9 
15.9 
15  9 
15  9 
15.9 
15  9 
15  9 
15.9 

15  9 

16  0 
16  0 
16  0 
16  0 
16  0 
16  0 
16  0 
16'0 
16  0 
16'0 
16  1 
16  1 
16  1 


16.1 

16.1 

16.1 

16.1 

16.1 

16.1 

16.1 

16.2 

16.2 

16.2 

16.2 

16.2 

16.2 

16.2 

16.2 

16.2 

16.2 

16.3 

16.3 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16.4 

16.4 

16.4 

16.4 

16.4 

16.4 

16,4 

16.4 


Tan<2 


D.  10"!  Cotang.  jjN.sine.jN.  t-os 


9.877114 
877377  ^^' 
877640^^- 
877903  I ^^' 
878165^^' 

.  878428  2^ • 

878691  ;::  • 

878953  K:;  • 
879216  I  ^• 
879478 'TX- 
879741  ^ 


9.88fX)03 
880265 
880528 
880790 
881052 
881314 
881576 
881839 
882101 
882363 

9.882625 
882887 
883148 
883410 
883672 
883934 
884196 
884457 
884/19 
884980 

9.885242 
885503 
885/65 
886026 
886288 
886549 
886810 
887072 
88/333 
887594 

9.887^f5 
888116 
8883  77 
888639 ! 
888900 
889160 
889421 
889682 
889943 
890204 
890465 
890725 
890986 
891247 
891507 
891 768 
892028 
892289 
892549 
892810 


43, 
43, 
43, 
43 
43, 
43, 
43, 
43, 
43, 
43, 
43, 
43 
43 
43, 
43, 
43, 
43, 
43. 
43, 
43, 
43, 
43, 
43, 
43 
43 
43 
43, 
43, 
43, 
43, 
43, 
43, 
43, 
43, 
43, 
43, 
43, 
43, 
43, 
43, 
43, 
43, 
43, 
43. 
43. 
43, 
43. 
143. 
43. 
43. 


10 


10 


10. 


10 


10 


10 


122886 
122623 
122360 
122097 
121835 
121572 
121309 
121047 
120784 
120522 
120259 
119997 
119735 
119472 
119210 
118948 
118686 
118424 
118161 
117899 
117637 
117375 
117113 
116852 
116590 
116328 
116066 
115804 
115543 
115281 
115020 
114758 
114497 
114235 
113974 
113712 
113451 
113190 
112928 
112667 
112406 
112145 
111884 
111623 
111361 
111100 
110840 
110579 
110318 
110057 
109796 
10J535 
109275 
109014 
108753 
1  (>8493 
108232 
10-972 
107'Jn 
10/451 
10719U 


|Q«(^  !«fn«ol 


60182  79864 


60205 
60228 
60251 
60274 
60298 
60321 
60344 
60367 
: 60390 
160414 


79846 
79829 
79811 
79793 
79776 
79758 
79741 
79723 
79706. 
79688 


60437 179671 
604ti0  79658 


60483 
1 60506 
1 60529 
{ 60553 
160576 
! 60599 
1 60622 
1 60645 
' 60668 
160691 
,60714 
160738 
160761 
'60/84 
J6080/ 


60830 


! 60853 
608/6 
j  60699 
I 60J22 
! 60945 
i  60968 
160991 
161015 
,61038 
'61061 


79635 
79618 
79600 
79583 
79565 
79547 
79530 
79512 
79494 
79477 
79459 
79441 
79424 
79406 
79388 
79371 
79353 
79335 
79318 
79300 
79282 
/9264 
79247 
79229 
79211 
79193 


,61084, /9r76 

:6110/|79158 
61130179140 
6115379122 
61J  76  79105 
6119917908/ 
61222/9069 
61245  79051 
61268/9033 
612911/9016 
61314|<8998 
6133/78^80 
61 360;  78962 
61383178944 
61406  78926 
6142976908 
61451 
61474 
6149/ 

161520 

I f 1543 

i  eViio 

I  N  cn.s.  X.vine. 


76891 
78873 
8655 
8837 
6819 
8801 


60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 


52  Dc-grees. 


TABLK  II. 


Log.  Sines  and  Tangents.    (38°)    Natural  Sinw. 


5S 


DTTo^ 


D.  10")    Cotang.      N.sine.lN.  cos. 


ob. 789342  or 

1  789504  ;^ 

2  78J(>tio  ;"• 
789827  ;"• 
789988  ^■ 

790149  ;:!• 

791)bl0  *^ 

790471  f^ 

79J632  ^^• 

9   79J793  ,,'• 

JO   790954  -^• 

11  9.791115  ^^.• 

12  791275  ;;,^- 


13  791430  ^" 

14  ""'--'"^ 


791430  ;,  • 
791590  ^.^• 

15  J791757  ;^- 

16  •"""•"  ^^ 


791917  il 
17   792077  20 


792077  -" 

18  792237  .j^" 

19  792397  ^.' 

20  792557  ^^• 

21  9.792710  ;^- 

22  792870  -Tf/ 

23  - "-  ^^ 

24 


792870  - 

23  793035  „p 

24  793195  fy 

25  793354  ;;"• 
20   793514  ^^• 

27  793073  I  Zi 

28  793832 !  *^ 

29  793991  I  ;.^ 

30  794150  ^r 

31  9.794308,^^- 

32  794407  ;^- 
334.  794020  ;:p' 

34  794784^^- 

35  794942';)^" 
30   795101  ijp 

37  795259  ^^: 

38  795417  ^r 

39  795575  ^J^' 

40  795733  ^^" 

41  9.795891  g^" 

42  790049  ^J" 

43  -  790200  f^ 

44  79«J304  ;}i 

45  790621  ^^ 
790079  OK 
7»ii836  ^ 
790993  ^ 
797150  .':^ 


40 

47 
48 
49 
60 


52 
63 
64 
55 
50 
57 
58 
69 
00 


79/150    -^^. 

797307   ,,2 

61  9.797404  T^ 


Cosine. 

9,896532 
890433 
890335 
890230 
890137 
890038 
895939 
895840 
895741 
895041 
895542 
,89.5443 
895343 
895244 
895145 
895045 
894945 
894840 
894740 
894040 
894540 
.894440 
894340 
894240 
894140 
894040 
893940 
893840 
893745 
893045 
893544 
1.893444 
893343 
893243 
-893142 
893041 
892940 
892839 
892739 
892038 
892530 

9.892435 
892334 
892233 
892132 
892030 
891929 
891827 
891720 
891024 
891523 


.797404  ;" 

797021  ^ 

797777  1^. 

797934  20 

798091  OK 

798247  ;^ 

798403  il 

798500  - 

798710  ;^ 

798872  '^^ 


8 
.8 
8 
8 
8 
8 
7 
7 
7 
7 
7 
7 
6 
6 
0 
.0 
6 
G 
6 
5 
6 
5 
5 
6 
.4 
4 
4 
4 
4 
4 
A 
3 
3 
■  3 
3 
3 
3 
3 
2 
2 
2 
2 
2 
1 

;}j9. 891421 


Cosine. 


891319 
891217 
891115 
891013 
89091 1 
890809 
890707 
890005 
890503 


Sine. 


D.  10" 


16.4 
10.5 
10.5 
10.5 
10.5 
10.5 
10.6 
10.5 
10.6 
10.6 
10.5 
16.6 
16.0 
10.6 
10.0 
10.0 
10.0 
10.0 
16.6 
10.6 


10.6 

16.7 

16.7 

16.7 

10.7 

16.7 

16.7 

16.7 

16.7 

16.7 

16.7 

16.8 

16.8 

10.8 

10.8 

16.8 

10.8 

10.8 

10.8 

10.8 

10.8 

16.9 

10 

10 

10 

10 

10 

10 

10 

10 

17.0 

17.0 

17.0 

17.0 

17.0 

17.0 

17.0 

17.0 

17.0 

17.0 


Tang. 


.892810 
893070 
893331 
893591 
893851 
894111 
894371 
894632 
894892 
895152 
895412 

.895672 
895932 
890192 
890452 
890712 
890971 
897231 
897491 
897751 
898010 

.898270 
898530 
898789 
899049 
899308 
899508 
899827 
900080 
900340 
900005 

.900804 
901124 
901383 
901042 
901901 
902100 
902419 
902079 
902938 
903197 

.903455 
903714 
903973 
904232 
904491 
904750 
905008 
905207 
905520 

■  906784 

i.  900043 
900302 
900500 
900819 
907077 
907336 
907594 
907852 
908111 
908369 

Cotiuig. 


61772;78040 


61804 

01887 

01909 

61932 

61956J78490 

61978"'"-° 

62001 

62024 

62040 

02009  78405 
78387 
78309 
78351 
78333 
78315 
78297 
8279 
T8201 
78243 
78225 
;8200 
78188 
78170 

62388  78152 


02411 
02433 
62450 


78134 
78110 
78098 


62479  78079 
02502  78001 
02524  78043 
62547  78025 
02570  78007 
62592  77988 
02015  77970 
02038177952 


Tung. 


N.  COS.  N.«ine 


51  Dugre«a. 


60 


Log.  Sines  and  Tangents.    (39*^)    Natural  Sines. 


TABLE  IL 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

lU 

H 

12 

13 

14 

15 

IG 

17 

18 

I  19 
2U 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
3-2 
33 
34 
35 
3ti 
37 
38 
39 
40 
41 
42 
43 
44 
45 
40 
47 
48 
49 
60 
51 
52 
53 
54 
55 
50 
57 
58 
59 
00 


Sine. 


9.798772 
799028 
799184 
799339 
799495 
799051 
799806 
799902 
800117 
8002/2 
800427 
9. 800582 
800737 
800892 
801047 
801201 
801356 
801511 
801665 
801819 
801973 
9.802128 
802282 
802436 
802589 
802743 
8028y7 
803050 
803204 
803357 
80351 1 
9.803664 
803817 
803970 
804123 
804276 
804428 
804581 
804734 
804880 
805039 
9.805191 
805343 
805495 
805647 
805799 
805951 
806103 
806254 
80o406 
80J557 
9.806/09 
806860 
807011 
807163 
807314 
807465 
80/615 
807766 
807917 
808067 
Cosine. 


D.  10"|  Cosine. 


26.0 
26.0 
26.0 
25.9 
25.9 
25.9 
25.9 
25.9 
25.9 
25.8 
25.8 
25.8 
25.8 
25.8 
25.8 
25.8 
25.7 
25.7 
25.7 
25.7 
25.7 
25.7 
25.6 
25.6 
26.6 
25.6 
25.6 
25.6 
25.6 
25.5 
25.5 
25.6 


25 


25.5 
25.5 
25.4 
25.4 
25.4 
25.4 
25.4 
25.4 
25.4 
25.3 
25.3 
25.3 
25.3 
25.3 
25.3 
25.3 
25.2 
25.2 
.2 
25.2 
25.2 
25.2 
25.2 
25.1 
25.1 
25.1 
25.1 


9.890503 
890400 
890298 
890195 
89J0J3 
889990 
889888 
88978) 
-889682 
889579 
889477 

9.889374 
889271 
889168 
8S9064 
8&S961 
888858 
888755 
888651 
888548 
888444 
.888341 
888237 
888134 
888030 
887926 
887822 
887718 
887614 
887510 
887406 

9.887302 
887198 
887093 
886989 
88(>885 
886780 
886676 
886571 
886466 
886362 

9.886257 
886152 
886047 
885942 
885837 
885732 
885627 
885522 
885416 
885311 

9.8^5205 
885100 
884994 
88-1889 
884783 
884677 
884572 
884466 
884360 
884254 


D.  10" 


17.0 
17.1 
17.1 
17.1 
17.1 
17.1 
17.1 
17.1 
17. J 
17.1 
17.1 
17.2 
17.2 
17.2 
17.2 
17.2 
17.2 
17.2 
17.2 
17.2 
17.3 
17.3 
17.3 
17.3 
17.3 
17.3 
17.3 
17.3 
17.3 
17.3 
17.4 
17.4 


Tang. 


17 

17 

17 

17 

17 

17 

17.4 

17.4 

17.5 

17.5 

17.6 

17.5 

17 

17 

17 

17 

17 

17 

17 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 


9.903369 
90S;)28 
908886 
90J144 
90J4J2 
909660 
909918 
910177 
910435 
910 J93 
910951 
9.911209 
911467 
911724 
911982 
912240 
912498 
912756 
913014 
913271 
913529 
9.913787 
914044 
914302 
914560 
914817 
915075 
915332 
915590 
915847 
916104 
9.916362 
916619 
916877 
917134 
917391 
917648 
917905 
918163 
918420 
918677 
9.918934 
919191 
919448 
919705 
919962 
920219 
920476 
920733 
920990 
921247 
9.921503 
921760 
922017 
922274 
922530 
922787 
923044 
923300 
92355  7 
923813 
Co  tan-'. 


D.  10' 


43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42,9 

42.9 

42.8 

42.8 

42.8 

42.8 

42.8 

42,8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.7 


IN.  .siiic.)N.  cofl.l 


62932 


77715 


62955  77696 


10.091631 

091372 

091114 

090856 

090598 

090340 

090J82 

089823 

089565 

089307 

089049 
10.088791 

088533 

088276 

088018 

087760 

087502 

087244 

086986 

086729 

086471 
10.086213 

085956 

085698 

085440 

085183 

084925 

084668 

084410 

084153 

083896 
10-083638 

083381 

083123 

082866 

082609 

082352 

082095 

081837 

081580 

081323  !|  63832  769 
10  081066  1,63854 

0808091!  6387, 

080552 i  163899 

080295 j  I  63922 

0800381163944 

079781  11  63966  (7o866 

0795241  6398  j|76847 

079267  i!  6401 176828 


62977 
6300J 
63022 
63045 
63068 

1 63090 
63113 
63135 
63158 
93180 
63203 
63225 
63248 
63271 
63293 
63316 
63338 
63361 
63383 
63406 
63428 
63451 
63473 
63496 

163518 

163.540 
63563 

1 63585 
63608 


63630 
63653 
63675 
63698 
163720 
1 63 742 
!  6371)5 
! 63787 
,63810 


77678 
77660 
77641 
77623 
77605 
77586 
7751)8 
77550 
77631 
77513 
77494 
77476 
77458 
77439 
77421 
77402 
77384 
77366 
77347 
77329 
77310 
77292 
77273 
77255 
77236 
77218 
77199 
77181 

162 
77144 
77125 
77107 

0o8 
77070 
77051 

0J3 
77014 
76996 


76959 
7t>940 
76921 
76903 

76884 


079010 ''64033 
078753  1164056 
10.0784971164078 


078240' 

07/9831 

077726 

077470 

077213 

076956 

0/6700 

076443 

076187 


64100 
6412o 
64145 
6416  7 
64190 
04212 
64234 
64256 
64279 


Tan* 


N.  COS.  N.t^iiu 


,6810 
.6791 
76772 
76754 
76735 
76717 
7u698 
76679 
76661 
,6642 
76623 
i6604 


60 
69 
58 
67 
66 
55 
64 
53  i 
62. 
51 
60 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
6 
4 
3 
2 
1 
0 


50  Degrees. 


TAIiLli  11. 


l.o;;.  .>iiies  and  Tangents.    (40°)     Natural  Sines. 


61 


Of  J. 

1 

2 

3 

4 

5 

ti 

7l 

loi 

lll9 
V2 
13 
14 
15 
lo 
17 
18 
19 
20 
21 
22 
23 
24 
25 
21J 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
I  4ti 
47-1 
48 
49 
50 
51 
52 
63 
64 
65 
5(i 
57 
58 
59 
60 


Sine. 

80-iOJ7 
808il8 
80-}3.)8 
8a8jl9 
80.-iJ6'^ 
8j>-i«19 
8J89d9 
80J119 
8i)J2U9 
8JJ419 
80J569 
809718 
8l)98o8 
810017 

-8l01l)7 
810316 
810465 
810614 
811)763 
810912 
811061 

.S11210 
811358 
811507 
811655 
811804 
811952 
81210J 
812248 
812396 
812544 

.812692 
812840 
812988 
813135 
813-^83 
813430 
813578 
813725 
8138/2 
814019 

.814166 
814313 
814460 
814607 
8U763 
814900 
81504<) 
815193 
815339 
815485 

.815631 
815-78 
815924 
8160u9 
816215 
816361 
816507 
816652 
816798 
816943 
Cosine.  | 


D.  Wl    Coatne.  |D.  10"|  Tang, 


25.1 

25.1 

25.1 

25.0 

25.0 

25.0 

25.0 

25.0 

25.0 

24.9 

24.9 

24.9 

24.9 

24.9 

24  9 

24.8 

24.8 

24.8 

24.8 

24.8 

24.8 

24.8 

24.7 

24.7 

24.7 

24.7 

24.7 

24.7 

24.7 

24.6 

24.6 

24.6 

24.6 

24.6 

24.6 

24.6 

24.5 

24.5 

24.6 

24.6 

24.6 

24.6 

24.5 

24.4 

24.4 

24.4 

24.4 

24.4 

24.4 

24.4 

24.3  1, 

21.3 

24.3 

24.3 

24.3 

24.3 

24.3 

24.2 

24.2 

24.2 


.884254 

884148 

884042 

88393( 

883829 

883723 

883617 

883510 

883404 

88329; 

883191 
.883084 

882977 

882871 

882764 

88265/ 

8S2550 

882443 

882336 

882229 

882121 
.882014 

88190/ 

881799 

881692 

881584 

881477 

88l3«i9 

881261 

881153 

881046 
.880938 

880830 

880 ; 22 

880613 

880505 

8t;0397 

880-289 

880180 

8800/2 

879963 
.879855 

879746 

879637 

879629 

879420 

879311 

8792(J2 

879093 

878984 

8788/6 
.878766 

878G56 

878647 

878438 

878328 

878219 

878109 

877999 

877890  ,„  o 
_877780  ^^'^ 
"sine. 


17.7 
17.7 
.'J  17.7 
W.7 
17.7 
17.7 
17.7 
17.7 
17.7 
17.8 
17.8 
17.8 
17.8 
17.8 
17.8 
17.8 
17.8 
17.8 
17.9 
17.9 
17.9 
17.9 
[7.9 
17.9 
7.9 
17.9 
17.9 
17.9 
18.0 
18.0 
18.0 
18.0 
18.0 
18.0 
18.0 
18.0 
18.0 
18.1 
18.1 
18.1 
18.1 
18.1 
18.1 
18.1 
18.1 
18.1 
18.1 
18.2 
18.2 
18.2 
18.2 
18.2 
18.2 
18.2 
18.2 
18.2 
18.3 
18.3 
18.3 


.923813 
924070 
924327 
924583 
924840 
92509t) 
925352 
925«)09 
9258()5 
926122 
926378 

.926634 
926890 
927147 
927403 
927659 
927915 
928171 
928427 
928b83 
928940 

.929196 
929452 
929708 
929964 
930220 
930475 
930731 
930987 
931243 
931499 

••931755 
932010 
932266 
932522 
932778 
933033 
933289 
933545 
933800 
934056 

1.934311 
934567 
934823 
935078 
935333 
935689 
936844 
936100 
936355 
936610 

1.936866 
937121 
937376 
937632 
937887 
938142 
938398 
938()53 
938908 
939163 

Cotarig. 


D.  lO'^l  Cotang.  I  jN.Binc. 

'64279 
64301 


42.7 

42.7 

42.7 

42.7 

42.7 

42.7 

42.7 

42.7 

42.7 

42.7 

42.7 

42.7 

42.7 

42.7 

42 

42 

42 

42 

42 

42 

42.7 

42.7 

42,7 

42.7 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.5 

42,5 

42.6 

42.5 

42.6 


10.076187 
076930 
0756,3 
075417 
075160 
074904 
074648 
074391 
074136 
073878 
073622 

10.073366 
073110 
072853 
072597 
072341 
072085 
071829 
071573 
071317 
071060 

10.070804 
070548 
070292 
070036 
069780 
069525 
0ii92()9 
069013 
068767 
068501 

10.068245 
067990 
067734 
067478 
06.222 
066967 
066711 
066456 
066200 
0ti5944 

10.065689 
065433 
065177 
064922 
(I64()ti7 
064411 
064156 
063900 
063645 
063390 

10.0ti3134 
062879 
062624 
062368 
0O2113 
061858 
061602 
Ot)1347 
06109'2 
06083/ 


N.  COS. 


76604 
765>}6 
64323' 76567 
64346176548 
64308 1 76530 
64390  76511 


64412, 
64435 
64457 
64479 
64501 
64524 
64546 


76492 
76473 
76456 
7<)436 
76417 
76398 
76380 


645681 763ul 


1 64690 
164612 
I  (J4635 
164657 
1 64679 
i  64701 
1 64723 
j  04746 

64768 
164790 
164812 
I  64834 

64856 
1 64878 
164901 
i  64923 
i  64945 
j  64967 
i  (>4989 
I65U11 

65033 


7()342 
7()323 
7(io04 
7()286 
76267 
76248 
76229 
76210 
76192 
76173 
76154 
76135 
76116 
76097 
76078 
76059 
76041 
76022 
76UU3 
75984 
759u5 


65055] /594b 
|65077|769-^7 
165100175908 
I  65 1-^2 176889 
66144  75 ~^'0 
;5ft51 
75832 
75813 
76794 
76775 
/5«56 
76738 
75/19 
,6iOO 
/56b0 
/5061 
,  604VJ 
75623 
/5o04 
.5565 
/55o6 
7  554  / 
/  5528 
.5509 
/54b0 
/54/1 


'6oltj6 

65188 
165210 
i  66232 
:  65:^54 

65276 
'  65-iy8 

653  V  0 
1 65342 

65364 

()5b8o 

6540b 

65430 

65452 

65474 

65496 

6551b 

05540 

655<jVf 

655b  ^ 

65i>lk> 

N.  fos.  N.tsiue 


49  Degree*. 


62 


—] 


Log:.  Sines  and  Tangents.    (41°)    Natural  Sines. 


TABLE  n. 


2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
t8 
49 
50 
51 
62 
53 
54 
55 
56 
57 
58 
59 
60 


Sine. 

3.816943 
817088 
817233 
817379 
817524 
817668 
817813 
817958 
818103 
818247 
818392 

3.818536 
818681 
818825 
818969 
819113 
819257 
819401 
819545 
819689 
819832 

>. 819976 
826120 
820263 
826406 
82055U 
820693 
820836 
820979 
821122 
821265 

>. 821407 
821560 
821693 
821835 
821977 
822120 
822262 
822404 
822646 
822688 

1,822830 
822972 
823114 
823255 
823397 
823639 
8i3680 
823821 
823963 
824104 

1.824245 
824386 
824527 
824668 
8248  iJ8 
824949 
825090 
825230 
826371 
825511 

Cosine. 


D.  10"  Cosine. 


24.2 
24.2 
24.2 
24.2 
24.1 
24.1 
24.1 
24.1 
24.1 
24.1 
24.1 
24.0 
24.0 
24.0 
24.0 
24.0 
24.0 
24.0 
23.9 
23.9 
23.9 
23.9 
23.9 
23.9 
23.9 
23.8 
23.8 
23.8 
23.8 
23.8 
23.8 
23.8 
23.8 
23.7 
23.7 
23.7 
23.7 
23.7 
23.7 
23.7 
23.6 
23.6 
23.6 
23.6 
23.6 
23.6 
23.6 
23.6 
23.6 
23.6 
23.6 
23.6 
23.5 
23.5 
23.4 
23.4 
23.4 
23.4 
23.4 
23.4 


1.877780 
877670 
877560 
877450 
877340 
877230 
877120 
877010 
876899 
876789 
876678 
.876568 
876457 
876347- 
876236 
876125 
876014 
875904 
875793 
876682 
875571 
.875469 
875348 
875237 
875126 
875014 
874903 
874791 
874680 
874568 
874456 
.874344 
874232 
874121 
874009 
873896 
873784 
873672 
873560 
873448 
873335 
.873223 
873110 
872998 
872885 
872772 
872659 
872647 
872434 
872321 
872208 
.872096 
871981 
871868 
871765 
871641 
871528 
871414 
871o01 
871187 
871073 
Sine. 


D.  10' 


18.3 
18.3 
18.3 
18.3 
18.3 
18.4 
18.4 
18.4 
18.4 
18.4 
18.4 
18.4 
18.4 
18.4 
18.5 
18.5 
18.6 
18.5 
18.6 
18.5 
18.5 
18.5 
18.5 
18.6 
18.6 
18.6 
18.6 
18.6 
18.6 
18.6 
18.6 
18.6 
18.7 
18.7 
18.7 
18.7 
18.7 
18.7 
18.7 
18.7 
18.7 
18.7 
18.8 
18.8 
18.8 
18.8 
18.8 
18.8 
18.8 
18.8 
18.8 
18.9 
18.9 
18.9 
18.9 
18.9 
18.9 
18.9 
18.9 
18.9 


Tana;. 


3.939163 
939418 
939673 
939928 
940183 
940438 
940o94 
940949 
941204 
941458 
941714 

3,941968 
942223 
942478 
942733 
942988 
943243 
943498 
943752 
944007 
944262 

3.944617 
944771 
945026 
945281 
945535 
945790 
946045 
946299 
946554 
946808 

3.9470j3 
947318 
947672 
947826 
948081 
948336 
948590 
948844 
949099 
949363 

3.949607 
949862 
9501 16 
950370 
950625 
950879 
951133 
951388 
951642 
951896 

),95216U 
952405 
952659 
952913 
963167 
953421 
953676 
953929 
954183 
954437 
Cotanj'. 


D.  10' 


42.5 

42.5 

42.5 

42.6 

42.5 

42.6 

42.5 

42.6 

42.5 

42.6 

42.5 

42.5 

42.6 

42.5 

42  6 

42.5 

42.6 

4-2.6 

42.5 

42.5 

42.6 

42.6 

42 

42 

42 

42 

42 

42 

42 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.3 

42.3 

42.3 

42.3 

42.3 


Cctang.  I, N.  sine. 

10.030837  1 166606 
0605821!  65628 
060327  I  j 66650 


75471 
75452 
75433 


060072  165672  76414 
059817'!  66691 
059562;!  65716 
059306  ,65738 


75395 
75375 
75366 


059051  l65759|75337 


058796  65781 
058542 j  165803 
058286 '165825 

10.058032  1 1  65847 
05777711  65869 
057522!  I  65891 
057267  65913 
057012;!  65935 
056767  '65956 
056502  165978 
0562481  66000 
055993  I' 66022 
055 738!  166044 

10.055483;  66066 
055229  H  66088 
064974!  66109 
054719  I!  66131 
054465  1 166153 
054210  66175 
053955'  66197 
0537011  66218 
063446 '66240 
053192'  66262 

10.052937'  66284 
05J682  '  66306 
0524281166327 
0521741:66349 
051919  i|  66371 
051664'  66393 
051410  66414 
051156  66436 
050901  1  66458 
050647 '66480 

10.050393  I  66501 
050138  I  66523 
049884 !  66545 
0496301  665b6 
049376'  66588 
049121 1  66610 
0488671  66632 
0486121  66653 
0483581  66675 
048104'  66697 

10.047860  66718 
047696;  66740 
047341  '66762 


uo.. 

75318 

75299 

75280 

75261 

75241 

75222 

75203 

75184 

75165 

75146 

75126 

75107 

75088 

76069 

75050 

75030 

76011 

74992 

74973 

74953 

74934 

74915 

74896 

74876 

74857 

74838 

74818 

4799 

4780 

4760 

74741 

4722 

74703 

4683 

74663 

4644 

4625 

74606 

74586 

74567 

74548 

4522 

4509 

74489 

74470 

4451 


04708  7  1  66783174431 
046833  1166805 
0465  79  '66827 
046325'  66848 
046071  66870 
045817 -'66891 
045563'  66913 


Tanj^. 


74412 
74392 
74373 
74353 
74334 
74314 


60 

59 
68 
57 
56 
55 
54 
,63 
I  52 
61 
60 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
36 
34 
33 
32 
31 
30 
29 
28 
27 
26 
26 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10  ; 

8  1 

7  ' 

6 

5 

4 

3 

2 

1 

0 


48  Dej'rees. 


TA1JL1-:  II. 


Ix)g.  Sines  and  Tangents.     (42")     Natural  ciinc'S. 


63 


019 

I 

2 

3 

4 

6 

6 

7 

8 

9 
10 
11 
12 
13 
14 
13 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
2(i 
27 
28 
29 
30' 
31  9 
32 
33 


.825511 

825661 

825791 

825  ).l 

82(3  (71 

82t).ll 

82G];)! 

820491 

820031 

820  7.0 

820910 

8270.9 

837189 

8273-8  i 

8274i)7  I 

827600 

82 7; 45 

8278S4 

828023 

828102 

828301 
.828439 

828578 

828710 

828855 

828993 

829131 

82^209 

829407 

829545 

829683 
.829821 

829959 

830097 

830234  .  ;. 

830372  ^ 


830509 
830.)40 
830784 
830921 
831058 
.831195 
831332 
831409 


831o06  -; 
831742  ^^ 


22 
22 
22 
22 
22 
22 
22 
22 
22 

833105  22 

833241  t, 

833377  t; 

83  3512  ** 

833048  ,■;.'; 

833783  ^^ 
Oxsine. 


831879 
832U15 
832152 
83-2288 
832425 
832661 
832097 
832833 
83-2969 


9.871073 
870.>0iJ 
870840 
870732 
870618 
870501 
870^90 
870270 
870161 
8;0J47 
8  .99  }3 

9.869818 
-869704 
869589 
869474 
869360 
869245 
869130 
809015 
8)8900 
-868785 

9.868070 
868555 
808440 
8J«324 
808209 
808093 
867978 
867862 
867747 

^  807031 

19.867515 
807399 
807-283 
807167 
807051 
800935 
800819 
860703 
80()586 
866470 
860353 
800-237 
8061-20 
860004 
86.5887 
866770 
805()53 
865530 
865419 
865302 

19.865185 
805068 
804950 
864833 
804716 
864598 
8(>4481 
864303 
864245 
804127 
.Sine. 


19. 
19. 
19, 
19, 
19, 
19, 
19, 
19, 
19 
19, 
19, 
19, 
19, 
19 
19, 
19, 
19 
19 
19 
19 
19 
19 
19, 
19, 
19, 
19, 
19. 
19, 
19. 
19. 
19. 
19. 
19. 
19. 
19. 
19. 
19. 
19. 
19. 
19. 
19. 
19, 
19, 
19, 
19, 
19, 
19, 
19, 
19, 
19, 
19, 
19 
19, 
19, 
19, 
19, 
19 
19 
19 
19, 


'  Sine.      D.  10"     Co.sin«.     D.  10"(     Tang.      D.  10"     Cotung.     |,N.  sine. [N.  cos. 


.954437 
954691 
954945 
955-200 
955454 
955707 
955901 
950215 


42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 

950409  to'" 


950723 
950977 

.957231 
957485 
957739 
957993 
958246 
958600 
958754 
959008 
959202 
959516 

.959769 
900023 
960277 
900531 
900784 
961038 
961291 
961545 
901799 
902052 

.962300 
962560 
962813 
963067 
903320 
903574 
963827 
904081 
964335 
904588 

.904842 
965095 
905349 
905602 
965855 
900109 
9603(i2 
906616 
966869 
967123 

.967376 
967629 
967883 
968136 
968389 
968643 
968896 
969149 
969403 
969656 

Cotaug. 
47  Degreey. 


42.3 
42.3 
42.3 
42.3 
42.3 
42. 3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 
42.2 


10.045563 
045309 
045056 
044800 
044540 
044293 
0+1039 
043785 
043531 
043277 
043023 

10.042769 
(H2615li 
042-261  : 
042007 
041764 
041500 
041240 
040992; 
04U738 
040484 

10.040231: 
039977: 
0b9723l 
0394691 
03921611 
038962 
038709 
038455 
038201  1 1 
03794811 

10.037694 i 
037440 
037187 
03(i933 
036680 
030426 
036173  |l 
035919'. 
035666  i! 
035412 'i 

10.035158: 
034905 


6691374314 
6693574295 
66956 174276 
06978174256 
6699974237 
67021 '74217 
67043174198 
07064174178 
67086.74169 
67107174139 
67129  74120 
67151174100 
67172174080 
67194174061 
67215:74041 
67-237 1740-22 
6725874002 
6728073983 
67301173963 
673-23173944 
673441739-24 
67366  73904 
67387173885 
6V409'73865 
67430:73846 
67452173826 
67473'73806 
67495:73787 
67516:73767 
67538173747 
67659173728 
6768073708 


67602 
67623 
67645 
67666 


73688 
73669 
73649 
73()-29 


034e;51  " 
034o98 
034145 
033891 
033638 
033384 
033131 
03-2877:' 
10.03-26-24; 
032371 i 
0321 17 j 
0318641 
031611  I 
031357 
031104 
03U851 
030597 
030344 
Tang. 


67688173610 

67709  73590 

67730  7357  0 

67762J73551 

67773173531 

67796173511 

67816173491 

67837173472 

67859'73452 

6788073432 

67901 

67923 

67944 

67965 


i3413 
73393  j 
73373; 
73  53j 
67987173333; 


68U08 
68029 
68051 
680/2 
68093 
68115 
68136 
68157 
68179 
68200 


73314 
3294 
3274 
73264 
73234 
73215 
73195 
73176 
73155 
73136 


N.  io.«.  N.. sine. I 


64 


Log.  Sines  and  Tangents.    (43°)    Natural  Sines. 


TABLE  II. 


0 
1 
2 
3 
4 
5 
6 
7 
t 
9 
10 
11 
12 
13 
14 
15 
It) 
17 
18 
19 
20 
21 
22 
23 
24 
25 
2(j 
27 
28 
29 
30 
31 
32 
33 
34 
35 
3d 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
67 
58 
59 
60 


Slue. 


1.833783 

833919 

834054 

834189 

83  4325 

834460 

834595 

834730 

834865 

834999 

835134 
,835269 

«^35403 

835538 

835672 

8358 J7 

835941 

8360/5 

836209 

836343 

836477 
.836611 

836745 

8368/8 

837012 

837146 

837279 

837412 

837546 

837679 

837812 
.83794 J 

8380/8 

S3»211 

838344 

838477 

838610 

838  742 

8388-6 

839007 

839140 
.8392  72 

839404 

839536 

839668 

83980J 

839932 

840064 

840196 

840328 

840159 
9.84U591 

840722 

840854 

840985: 

84iii6l;rJ 

841247  1^ 
841378  I „j 
841509 
841640 
8417/1 
sin". 


D.  10" 


Co.siiie. 


9.864127 
864010 
863892 
863774 
863656 
863538 
863419 
863301 
863183 
863064 
862946 

9.862827 
862709 
862590 
862471 
862353 
862234 
862115 
861996 
861877 
861758 

3.861638 

-861519 
861400 
861280 
861161 
861041 
860922 
860802 
860682 
860562 

9.860442 
860322 
8602U2 
860082 
859962 
859842 
859721 
8596U1 
859480 
859360 

9.859239 
859119 
858998 
8588  7  7 
858756 
858635 
858514 
858393 
858272 
858151 

9.858029 
857908 
857786 
85  7665 
857543 
85/422 
857300 
857178 
857056 
85J934 
Sine. 


P.  10' 

19.6 
19.6 
19.7 
19.7 
19.7 
19.7 
19.7 
19.7 
19.7 
19.7 
19,8 
19.8 
19.8 
19.8 
19.8 
19.8 
19.8 
19.8 
19.8 
19.8 
19.9 
19.9 
19.9 
19.9 
19.9 
19.9 
19.9 
19.9 
19.9 
20.0 
20.0 
20.0 
20.0 
20.0 
20.0 
20.0 
20.0 
20.1 
20.1 
20.1 
20.1 
2U.1 
20.1 
20.1 
20.1 
20.2 
20.2 
20.2 
20.2 
20.2 
20.2 
20.2 
2U,2 
20. 2 
20.3 
20.3 
20,3 
20.3 
20.3 
20.3 


Tang. 

9.969656 
969909 
970162 
970416 
970669 
970922 
971175 
971429 
971682 
971935 
972188 

9.972441 
972694 
972948 
973201 
973454 
973707 
973960 
974213 
974466 
974719 
974973 
975226 
976479 
975732 
975985 
976238 
976491 
976744 
976997 
977250 

9.977503 
977756 
978009 
978262 
978515 
978768 
979021 
979274 
979527 
979780 
.980033 
980286 
980538 
980791 
981044 
981297 
981550 
981803 
982056 
982309 

9.982562 
982814 
983067 
9J3320 
983573 
983826 
984079 
984331 
984584 
98483; 
Cotaii'j;. 


D.  10" 


42.2 

42.2 

42.2 

42,2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42,2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 


Cotaug.  iJ.V  .siuc.|N. 


10.030344 
030091 
029838 
029584 
029331 
029078 
028825 
028571 
028318 
028065 
027812 

10.027559 
027306 
027052 
026799 
026546 
026293 
026040 
025787 
025534 
025281 

10.025027 
024774 
024521 
024268 
024015 
023762 
023509 
023256 
023003 
022750 

10.022497 
022244 
021991 
021738 
021485 
021232 
020979 
020726 
020473 
020220 

10.019967 
019714 
019462 
019209 
018956 
018703 
018450 
018197 
017944 
017691 

10.017438 
017186 
016933 
016680 
016427 
016174 
015921 
015669 
015416 
115163 
Tanij. 


168200  73135 


68221 
i  68242 
1 68264 
168285 
1 68306 

68327 


73116 
73096 
73076 
73056 
73036 
73016 


!  1 68349172996 

!  1 683  70172976 

'68391  72957 

I  68412172937 

68434172917 

i  1 68455172897 

i!684/6'72877 

1 1 6849  7 172857 

j  68518172837 

j]  68539  72817 

jj  68561  72797 

168582  72777 

'168603  72757 

68624  72737 

68645  72717 

68666172697 

68688 172677 

68709172657 

68730r72637 

68751|72617 

68772'72597 

68793:725  77 

68814|72557 

68835 172537 

68857j72517 

68878172497 

68899  72477 

68920  72457 

68941 172437 

68962|7241 7 

68983|72397 

:  1 69004172377 

I  69025^72357 
1169016  72337 
|i69067!72317 

|69088!72297 
i  I  6910972277 
1169130  72257 
'6915172236 
;!  69172  72216 
6^)193  72196 
■69214  72176 
{69235  72156 
169256  72136 
16927/172116 
'169298  72096 
69319  72075 
I  69340 172055 
i!  69361 -72035 
:  1 69382  7201 5 

II  69403  71995 
69424  71974 

i  69445  7 1951 
!  69466^71931 


N.  ous.iN. 


46  Degrees. 


TABLE  II. 


Log.  Sines  and  Tangents.    (U°)    Natural  Sine*. 


65 


u 

9.841771 

1 

841902 

•2 

S42033 

;j 

8421(i3 

4 

842294 

6 

842424 

G 

842555 

7 

842.iH5 

8 

842815 

9 

84294b 

lU 

84307b 

11 

9.84320(i 

1-2 

843336 

13 

8434bb 

14 

84;J696 

16 

843/25 

lb 

843855 

17 

843984 

18 

844114 

19 

844243 

20 

844372 

21 

9.844502 

22 

844631 

23 

844760 

24 

844889 

26 

845018 

2G 

845147 

27 

845276 

28 

845405 

29 

845533 

30 

845662 

31 

9.845790 

32 

845919 

33 

846047 

34 

846175 

35 

846304 

3G 

846432 

37 

846560 

38 

846688 

39 

846816 

40 

84*i944 

41 

9.847071 

42 

847199 

43 

847327 

44 

847454 

45 

847582 

Hi 

847709 

47 

847836 

48 

84/964 

49 

8480JI 

60 

848218 

5! 

9.848345 

62 

848472 

63 

848599 

64 

848  r26 

65 

H  I8tt52 

66 

8489.9 

67 

849106 

58 

819232 

69 

849359 

60^ 

849485 

Cosine. 


Cotang.     I  N.  sino.  N.  cos. 


10.0151G3 
0'4910 
014657 
014104 
014152 
013899 
013646 
013393 
013140 
012888 
012635 

10.012382 
012129 
011877 
011b24 
011371 
011118 
010866 
010613 
010360 
010107 

10.0U9855 
009602 
009349 
009097 
008844 
0118591 
■  ^338 
0u<->08b 
007833 
007580 

10  007328 
00.0/5' 
006822  I 
006570 i 
006317 i 
00o064! 
005811) 
005559 ' 
005306  j 
005053 

10- 004801  I 
004548 ! 
004295 
004043 ' 
0J3.90 
003537  i 
003285 
003032 
002779 
002527 ; 

10  002274 
002021 ■ 
001769 
00151b  I 
001263 
001011  i 
000758  ! 
000505, 
00i>253!' 
000000  I 


6;>4()t) 
6948 
'>95US 
; 69529 
69549 
69570 
()9591 
69(il2 
691533 
69654 
69676 
69()9( 
69il1 
69737 
: 69 758 
69779 
6980.) 
69821 
69842 
6986' 
69o83 
69901 
69925 
69946 
()99()6 
69987 
70008 
700-29 
70049 
700/0 
70091 
70112 
7013 
70103 
70174 
70195 
70215 
70236 
70257 
702/7 
70298 
70319 
70339 
703()0 
70..81 
70401 
70422 
70443 
70463 
70484 
70505 
70525 
70546 
7056  < 
70587 
70(i08 
70628 
70649 
70670 
70690 
70711 


71934 
71914 
71894 
71873 
718i,3 
71833 
71813 
71792 
71772 
71752 
71732 
71711 
71691 
71671 
71650 
71630 
71610 
71590 
71569 
71549 
71529 
71508 
71488 


71468  137 


71447 
71427 
71407 
71386 
71366 
71345 
/1 325 
71305 
71284 
71264 
71243 
71223 
71203 
71182 
71162 
71141 
/1 121 
71100 
1080 


Tang.      11  N.  coc.  N.ciin 


1059  17 
/1038j  16 
71019' 16 
70998  14 
709/8 
70967 

0937 
70916 
7LI896 

0876 
70855 
70834 
70813 
70793 
/0772 
i0752 
70731 
/O.ll 


45  Degrees. 


36 

LOGARITHMS 

TABLE  III. 

LOGARITHMS   OF    NUMBERS. 

From  I  to  200, 

INCLUDING 

TWELVE  DECIMAL 

PLACES.    ( 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 

000000  000000 

41 

612783  856720 

81 

908485  018879 

2 

301029  995664 

42 

623249  290398 

82 

913813  852384 

3 

477121  254720 

43 

633468  455580 

83 

919078  092376 

4 

60-2039  991328 

44 

643452  676486 

84 

924279  286062 

6 

698970  004336 

45 

653212  513775 

85 

929418  925714 

6 

778151  250384 

46 

662757  831682 

86 

934498  451244 

7 

845098  040014 

47 

672097  857926 

87 

939519  252619 

8 

903089  98()992 

48 

681241  237376 

88 

944482  672150 

9 

954242  609439 

49 

690196  080028 

89 

949390  006645 

10 

yame  as  to  1. 

50 

Samo  as  to  5. 

90 

Same  as  to  9. 

11 

041392  685158 

51 

707570  176098 

91 

959041  392321 

12 

079181  246048 

62 

716003  343635 

92 

963787  827346 

13 

113943  352307 

63 

724275  8()9n01 

93 

968482  948554 

14 

146128  035678 

54 

732393  759823 

94 

973127  853600 

16 

176091  269056 

55 

740362  689494 

96 

977723  605889 

16 

204119  982656 

56 

748188  027006 

96 

982271  233040 

17 

230448  921378 

57 

755874  855672 

97 

986771  734266 

18 

255272  505103 

68 

763427  993563 

98 

991226  075692 

19 

278:53  600953 

69 

770852  011642 

99 

995635  194598 

20 

Same  as  to  2. 

60 

Same  as  to  6 

100 

Same  as  to  10, 

21 

322219  2947 

61 

785329  835011 

101 

004321  373783 

22 

342422  680822 

62 

792391  699498 

102 

008600  171762 

23 

,  361727  836018 

63 

799340  549453 

103 

012837  224705 

24 

380211  241712 

64 

806179  973984 

104 

017033  339299 

25 

397940  008672 

65 

812913  356643 

105 

021189  299070 

26 

414973  347971 

66 

819543  935542 

103 

025305  865265 

27 

431363  764159 

67 

826074  802701 

107 

029383  777685 

28 

447158  031342 

68 

832508  912706 

108 

033  423  755487 

29 

462397  997899 

69 

838849  090737 

109 

037426  497941    1 

30 

8ime  as  to  3. 

70 

Same  as  to  7. 

110 

Same  a*  to  11.     | 

31 

491361  693834 

71 

851258  348719 

111 

045322  978787    \ 

32 

505149  978320 

72 

857332  496431 

112 

049218  022670 

33 

518513  939878 

73 

863322  8G0120 

113 

053078  4434H3 

34 

631478  917042 

74 

869231  719731  ' 

114 

056904  851336 

36 

544068  044350 

75 

875 J61  263392 

115 

060397  840354 

36 

556302  500767 

76 

880813  692281 

116 

064457  989227 

37 

568201  724!)67 

77 

886490  725172 

117 

068185  861746 

38 

579783  596617 

78 

892094  602690 

118 

071882  007306 

39 

691064  607026 

79 

897627  091290 

119 

075546  961393 

^    40 

Same  a«  to  4. 

80 

Same  as  to  8. 

120 

Same  a^  to  12. 

• 

-  ^ 

OF  NUMBERS. 

67 

N. 

121 

082786  370316 

,  N. 

Log. 

N. 

Log 

148 

170261  715395 

176 

243038  048686 

122 

086359  830675 

149 

173186  268412 

176 

245612  1 1678 14 

123 

0b990t  111439 

;  150 

176091  259056 

177 

241973  2(i6362 

124 

093421  685162 

151 

178976  947293 

178 

250420  002309 

125 

096910  01 3008 

1  152 

181843  687945 

179 

252863  030980 

126 

100370  645118 

'  163 

184691  430818 

180 

256272  505103 

127 

103803  7-20956 

1  154 

187520  710836 

181 

257678  574^69 

128 

10/209  969(i48 

155 

190331  698170 

182 

2(i0071  387<.85 

129 

110589  710-299 

166 

193124  588354 

183 

262451  089730 

130 

Same  as  to  13. 

[  157 

195899  662409 

184 

264817  823010 

131 

117271  295656 

158 

198657  086954 

186 

267171  728403 

132 

120573  931-206 

159 

201397  124320 

186 

2(i9512  944218 

133 

123851  64091)7 

160 

204119  982656 

187 

271841  60fi536 

134 

127104  798365 

161 

206826  876032 

188 

274157  849264 

136 

130333  768495 

162 

209516  014543 

189 

276461  804173 

136 

133538  908370 

163 

212187  604404 

190 

278753  600953 

137 

136720  567156 

164 

214843  848048 

191 

281033  367248 

138 

139879  086401 

166 

217483  944214 

192 

283301  228704 

139 

143014  800-254 

166 

220108  088040 

193 

285557  309008 

140 

146128  035678 

167 

222716  471148 

194 

287801  729930 

141 

149219  112655 

168 

225309  281726 

195 

290034  611362 

142 

152288  344383 

169 

227886  704614 

196 

292256  071356 

143 

155336  037465 

170 

230448  921378 

197 

2944(;6  226162 

144 

158362  49-2095 

171 

232996  110392 

198 

296666  1902<j2 

145 

161368  002235 

172 

236528  446908 

199 

298863  07641 u 

146 

164352  855784 

173 

238046  103129 

147 

167317  334748 

174 

240549  248283 

LOG 

ARITHMS 

] 

OF 

•"ROM 

THE  PRI^ 
200  TO  1543, 

IE  ]^ 

lUMBERS 

IN 

CLUDING 

rwE 

LVE  DECIM 

AL  I 

'LACES. 

N. 
201 

Log. 

N. 
277 

Log. 

N. 

Log. 

303196  057420 

"442479  769064  i 

379 

578()39  209968 

203 

307496  037913 

281 

448706  319905 

383 

5831JJ8  773968 

207 

315970  345457 

283 

451786  435524 

389 

589949  (i01326 

209 

320146  286111 

293 

466867  620354 

397 

598790  50;i763 

211 

324282  455298 

307 

487138  375477  j 

401 

603144  372620   1 

223 

348304  863048 

311 

492760  389027 

409 

611723  308007   ' 

227 

356025  857193 

313 

495544  337546  , 

419 

622214  U22966   j 

229 

359835  482340 

317 

601059  262218  , 

421 

624282  095836   1 

233 

367356  92UK26 

331 

519827  993776 

431 

634477  270161   • 

239 

378397  900948 

337 

6276-29  900871 

433 

636487  89(>363 

241 

3820' 7  042575 

347 

640329  474791 

439 

642424  520242 

261 

399f»73  721481 

349 

642826  4-26959  ' 

443 

64()403  72(i223 

267 

409P33  123331 

353 

47774  705388 

449 

65-2246  341003 

263 

419955  748490 

359 

565094  448578 

457 

669916  200070   1 

269 

429  ?52  2800 J-2 

367 

664666  064252  , 

461 

663:00  9-25390 

271 

4?;2969  290874 

373 

571708  831809 

463 

665580  991018 

68 


LOGARITHMS 


N. 

Log. 

N. 

Loi?. 

N. 
1171 
1181 
1187 
11J3 
1  1201 

Log. 

467 
479 
487 
491 
499 

6i9.Jilu  580506 
68033')  513414 
68/(:28  961215 
691081  492123 
69ril00  545623 

821 
823 
827 
829 
839 

914343  157119 
915399  835212 
917505  50c)553 
918554  530550 
923761  960829 

06bbi6   896073 
0/2249  807613 
0744-0  718955 
076640  443670 
0.9543  007385 

503 
509 
521 
523 
641 

701 507  985056 
706717  782337 
716837  '/ 23300 
718501  688867 
733197  2o5107 

853 
857 
859 
863 
877 

930949  031168 
932980  821923 
938993  163831 
936010  795715 
942999  593356 

1213 
1217 
1223 
1229 
1231 

0838  30  800845 
085290  578-210 
087426  458017 
089551  882866 
090258  052912 

547 
557 
563 
569 
671 

737987  326333 
745855  195174 
75U5U8  394851 
755112  26(;393 
766636  1082io 

881 
883 
887 
907 
911 

944975  908412 
945960  703578 
947923  619832 
957607  287060 
959518  376973 

1237 
1249 
1259 

1277 
1279 

092369  699609 
096662  438356 
100026  729204 
103190  896808 
1068/0  642460 

577 
587 
593 
599 
601 

761175  813156 
768638  101248 
773U54  693364 
777426  8-^2389 
778874  472002 

919 
929 
937 
941 
947 

963315  511386 
968015  713994 
971739  590888 
973589  623427 
976349  979003 

1283 
1-.89 
1291 
1297 
1301 

108226  656362 
110252  917337 
110926  242517 
112939  986(K)6 
114277  296540 

607 
613 
617 
619 
631 

783138  6910^5 
787460  474518 
790285  164033 
791690  649020 
80J0-29  359244 

963 

967 

9;i 

977 
983 

979092  900638 
985426  474083 
987219  229908 
989894  563719 
992553  517832 

1303 
1307 
1319 
1321 

1327 

114944  415712 
116276  687564 
120244  795568 
120902  817604 
122870  922849 

641 
643 
647 
653 
659 

806858  f)29519 
808210  972924 
8109J4  280569 
814913  181275 
818885  414594 

991 

997 

1009 

1013 

1019 

996073  654485 
998695  158312 
003891  166237 
005609  445360 
008174  184006 

1361 
1367 
1373 
1381 
1399 

133858  125188 
135768  614554 
137670  537223 
140193  678544 
145817  714122 

661 
673 
677 
683 
691 

810201  459486 
828015  064224 
830588  668685 
834420  703682 
839478  047374 

1021 
1031 
1033 
1039 
1049 

009025  742087 
U 13258  665284 
014100  321520 
016615  547557 
020775  488194 

1409 
1423 
1427 
1429 
1433 

148910  994096 
153204  896557 
154424  012366 
155032  228774 
156246  402184 

701 

709 
719 
727 
733 

845718  017967 
850646  235183 
856728  89U383 
861534  410859 
865103  974742 

1051 
1061 
1063 
1069 
1087 

021602  716028 
025715  383901 
026533  264523 
028977  705209 
036229  544086 

1439 
1447 
1451 
1453 
1459 

158060  793919 
160468  531109 
161667  412427 
162265  614286 
164055  291883 

739 
743 
751 
767 
761 

868644  488395 
870988  813761 
855639  937004 
879095  879500 
881384  656771 

1091 
1093 
1097 
1103 
1109 

037824  750588 
038620  161950 
040206  627575 
042595  512440 
044931  546119 

1471 

1481 
1483 
1487 
1489 

167612  672629 
170555  058512 
171141  151014 
172310  968489 
172894  731332 

769 
773 

787 
797 
809 

885926  339S01 
888179  493918 
895974  732359 
901458  321396 
907948  521612 

1117 
1123 
1129 
1151 
1153 

018053  173116 
050379  756261 
052693  941925 
061075  323630 
061829  307295 

1493 
1499 
1611 
1523 
1531 

174059  807708 
175801  632866 
179264  464329 
182699  903324 
184976  190807 

811 

909020  854211 

1163 

066679  714728 

1643 

188366  926063 

OF    NUMBERS 


69 


AUXILIARY    LOGARITHMS 


N. 


,Oity 

.008 
,007 
008 
005 

1.003 
1.002 
1.001 


Log.                 ;l      N.        1 

003891 1()()237    - 

1.0)J9 

003400532110 

1.0003 

003029470554 

1.0)07 

00259S08li(Jb5 

i.oooy 

0U21U(j0fil76() 

M 

1.0005 

001-33712775 

1.0004 

001300933020 

1.0003 

0i)0.S«)7721529 

1.0u02 

000434077479   J 

l.OUOl 

Log. 


00039l)»i89248 
0i)034:29()(84 
000';03899,84 
00j2(J0^9.^547 
OO.J-jU  0929,0 
000I73(i83067 
0001302()8804 
00008085,  »2 11 
0U0;J434'J7277 


N. 


Log. 


1.00009 
1.00008 
1.00007 
1.00006 
1.00005 
1.00004 
1.00003 
1.00002 
1.00001 


000039083266 
00003474tH)91 
0(X)030398072 
00002()0.55410 
000021712704 
00;)017371430 
000013028638 
000008()85802 
000004342923 


N. 


1.000009 

1 .  oouo.js 

1.000007 
1.000006 
1 . 000005 
1.000004 
1.000003 
1.000002 
1.000001 


Log. 


000003908(528 
000003474338 
000003040047 
00000-'(i05756 
0000021 7 14(J4 
000001737173 
000001302880 
OU00008G8587 
000000434294 


1.0000001 
1.00000001 
[.  00001)0001 
, 000000 JOOl 


Log- 


000000043429 
000000004343 
000000000434 
000000000043 


(n) 
(o) 
(P) 

(q) 


m=0.43429448l9        log.  —1.637784298. 

By  the  preceding  tables  —  and  the  auxiliaries  A,  B,  and 
C,  we  can  tind  the  logarithm  of  any  number,  true  to  at  leant 
ten  decimal  places. 

But  some  may  prefer  to  use  the  following  direct  formula, 
which  may  be  found  in  any  of  the  standard  works  on  algebra: 


Log.  (z-|-l)=log.2r+0.8685889638/^_L  \ 


if  z  be 


The  result  will  be  true  to  twelve  decimal  places, 
over  2000. 

The  log.  of  composite  numbers  can  be  determined  by  the 
combination  of  logarithms,  already  in  the  table,  and  the  prime 
numbers  from  the  formula. 

Thus,  the  number  5083  is  a  prime  number,  find  its  log'a- 
rithm. 

We  first  find  the  log.  of  the  number  3082.  By  factoring, 
we  discover  thai  this  is  the  product  of  46  into  67. 


70  NUMBERS 


Log.  46,  1.6627578316 

Log.  67,  1.8260748027 

Log.  3082  3.4888326343 

Log.3083=3.4888326343+«-^^^^^^96'^« 


6165 


NUMBERS  AND  THEIR  LOGARITHMS, 

OFTEN    USED    IN    COMPUTATIONS. 

Circamference  of  a  circle  to  dia.  1 )  Log. 

Surface  of  a  sphere  to  dmmeter  IV  =3.14159265  0.4971499 
Area  of  a  circle  to  radiun  1  ) 

Area  of  a  circle  to  diameter  1  -—   .7853982  —1.8950899 

Capacity  of  a  sphere  to  diameter  1  =  .6236988—1.7189986 
Capacity  of  a  sphere  to  radius  1     =4.1887902       0.6220886 

Arc  of  any  circle  equal  to  the  radius  =57°29578  1.7581226 
Arc  equal  to  radius  expressed  in  sec.  =:206264"8  5.3144251 
Length  of  a  degree,  (radius  unity)  =  .01745329  —2.2418773 

12  hours  expressed  in  seconds,      =    43200  4.6354837 

Complement  of  the  same,       =0.00002315  — 5.3645163 

360  degrees  expressed  in  seconds,  =   1 296000  6. 1 1 26050 

A  gallon  of  distilled  water,  when  the  temperature  is  62° 
Fahrenheit,  and  Barometer  30  inches,  is  ^ILf^-J^j  cubic 
inches. 


^277.274=16.651542  nearly. 


277.274 


.775398 


=  1 8.78925284  ^  231  =  1 5. 1 98684. 


V  282  =16.792855. 


^".785398 

The  French  Metre=3.2808992,  English  feet  linear  mea- 
sure, =39.3707904  inches,  the  length  of  a  pendulum  vi- 
brating seconds. 


^^^-  _=  18.948708. 


VB  35967 


924227 


^A  53/ 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


